cohomology - maths.ed.ac.ukv1ranick/papers/topstrat.pdf · success in expanding to stratiﬁed...

The occurrence of singularities is pervasive in many problems in topology,

differential geometry and algebraic geometry. This book concerns the study

of singular spaces using techniques from a variety of areas of geometry and

topology and interactions among them. It contains more than a dozen expos-

itory papers on topics ranging from intersection homology, L2 cohomology

and differential operators, to the topology of algebraic varieties, signatures

and characteristic classes, mixed Hodge theory, and elliptic genera of singular

complex and real algebraic varieties. The book concludes with a list of open

problems.

Mathematical Sciences Research Institute

Publications

58

Topology of Stratified Spaces

Mathematical Sciences Research Institute Publications

1 Freed/Uhlenbeck: Instantons and Four-Manifolds, second edition2 Chern (ed.): Seminar on Nonlinear Partial Differential Equations

3 Lepowsky/Mandelstam/Singer (eds.): Vertex Operators in Mathematics and Physics

4 Kac (ed.): Infinite Dimensional Groups with Applications

5 Blackadar: K-Theory for Operator Algebras, second edition6 Moore (ed.): Group Representations, Ergodic Theory, Operator Algebras, and

Mathematical Physics

7 Chorin/Majda (eds.): Wave Motion: Theory, Modelling, and Computation

8 Gersten (ed.): Essays in Group Theory

9 Moore/Schochet: Global Analysis on Foliated Spaces, second edition10–11 Drasin/Earle/Gehring/Kra/Marden (eds.): Holomorphic Functions and Moduli

12–13 Ni/Peletier/Serrin (eds.): Nonlinear Diffusion Equations and Their Equilibrium States

14 Goodman/de la Harpe/Jones: Coxeter Graphs and Towers of Algebras

15 Hochster/Huneke/Sally (eds.): Commutative Algebra

16 Ihara/Ribet/Serre (eds.): Galois Groups over Q

17 Concus/Finn/Hoffman (eds.): Geometric Analysis and Computer Graphics

18 Bryant/Chern/Gardner/Goldschmidt/Griffiths: Exterior Differential Systems

19 Alperin (ed.): Arboreal Group Theory

20 Dazord/Weinstein (eds.): Symplectic Geometry, Groupoids, and Integrable Systems

21 Moschovakis (ed.): Logic from Computer Science

22 Ratiu (ed.): The Geometry of Hamiltonian Systems

23 Baumslag/Miller (eds.): Algorithms and Classification in Combinatorial Group Theory

24 Montgomery/Small (eds.): Noncommutative Rings

25 Akbulut/King: Topology of Real Algebraic Sets

26 Judah/Just/Woodin (eds.): Set Theory of the Continuum

27 Carlsson/Cohen/Hsiang/Jones (eds.): Algebraic Topology and Its Applications

28 Clemens/Kollar (eds.): Current Topics in Complex Algebraic Geometry

29 Nowakowski (ed.): Games of No Chance

30 Grove/Petersen (eds.): Comparison Geometry

31 Levy (ed.): Flavors of Geometry

32 Cecil/Chern (eds.): Tight and Taut Submanifolds

33 Axler/McCarthy/Sarason (eds.): Holomorphic Spaces

34 Ball/Milman (eds.): Convex Geometric Analysis

35 Levy (ed.): The Eightfold Way

36 Gavosto/Krantz/McCallum (eds.): Contemporary Issues in Mathematics Education

37 Schneider/Siu (eds.): Several Complex Variables

38 Billera/Bjorner/Green/Simion/Stanley (eds.): New Perspectives in Geometric

Combinatorics

39 Haskell/Pillay/Steinhorn (eds.): Model Theory, Algebra, and Geometry

40 Bleher/Its (eds.): Random Matrix Models and Their Applications

41 Schneps (ed.): Galois Groups and Fundamental Groups

42 Nowakowski (ed.): More Games of No Chance

43 Montgomery/Schneider (eds.): New Directions in Hopf Algebras

44 Buhler/Stevenhagen (eds.): Algorithmic Number Theory: Lattices, Number Fields,

Curves and Cryptography

45 Jensen/Ledet/Yui: Generic Polynomials: Constructive Aspects of the Inverse Galois

Problem

46 Rockmore/Healy (eds.): Modern Signal Processing

47 Uhlmann (ed.): Inside Out: Inverse Problems and Applications

48 Gross/Kotiuga: Electromagnetic Theory and Computation: A Topological Approach

49 Darmon/Zhang (eds.): Heegner Points and Rankin L-Series

50 Bao/Bryant/Chern/Shen (eds.): A Sampler of Riemann–Finsler Geometry

51 Avramov/Green/Huneke/Smith/Sturmfels (eds.): Trends in Commutative Algebra

52 Goodman/Pach/Welzl (eds.): Combinatorial and Computational Geometry

53 Schoenfeld (ed.): Assessing Mathematical Proficiency

54 Hasselblatt (ed.): Dynamics, Ergodic Theory, and Geometry

55 Pinsky/Birnir (eds.): Probability, Geometry and Integrable Systems

56 Albert/Nowakowski (eds.): Games of No Chance 3

57 Kirsten/Williams (eds.): A Window into Zeta and Modular Physics

58 Friedman/Hunsicker/Libgober/Maxim (eds.): Topology of Stratified Spaces

Volumes 1–4, 6–8, and 10–27 are published by Springer-Verlag

Topology of

Stratified Spaces

Edited by

Greg Friedman

Texas Christian University

Eugenie Hunsicker

Loughborough University

Anatoly Libgober

University of Illinois at Chicago

Laurentiu Maxim

University of Wisconsin-Madison

Greg FriedmanDepartment of MathematicsTexas Christian University

[email protected]

Anatoly LibgoberDepartment of Mathematics

University of Illinois at [email protected]

Eugenie HunsickerDepartment of Mathematical Sciences

Loughborough [email protected]

Laurentiu MaximDepartment of Mathematics

University of [email protected]

Silvio Levy (Series Editor)Mathematical Sciences Research Institute

Berkeley, CA [email protected]

The Mathematical Sciences Research Institute wishes to acknowledge support bythe National Science Foundation and the Pacific Journal of Mathematics for the

publication of this series.

cambridge university press

Cambridge, New York, Melbourne, Madrid, Cape Town,Singapore, Sao Paulo, Delhi, Tokyo, Mexico City

Cambridge University Press32 Avenue of the Americas, New York, NY 10013-2473, USA

www.cambridge.orgInformation on this title: www.cambridge.org/9780521191678

c© Mathematical Sciences Research Institute 2011

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2011

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Topology of stratified spaces / edited by Greg Friedman ... [et al.].p. cm. – (Mathematical Sciences Research Institute publications ; 58)

Includes bibliographical references.ISBN 978-0-521-19167-8 (hardback)1. Singularities (Mathematics) 2. Algebraic spaces. 3. Algebraic topology.

I. Friedman, Greg, 1973–

QA614.58.T67 2011516.3′5—dc22 2010051108

ISBN 978-0-521-19167-8 hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLsfor external or third-party Internet Web sites referred to in this publication and does notguarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Topology of Stratified SpacesMSRI PublicationsVolume58, 2011

Contents

Preface ix

An introduction toL2 cohomology 1X IANZHE DAI

The almost closed range condition 13GILLES CARRON

Rigidity of differential operators and Chern numbers of singular varieties 31ROBERT WAELDER

Hodge theory meets the minimal model program: a survey of logcanonical anddu Bois singularities 51

SANDOR J. KOVACS AND KARL SCHWEDE

Elliptic genera, real algebraic varieties and quasi-Jacobi forms 95ANATOLY L IBGOBER

The weight filtration for real algebraic varieties 121CLINT MCCRORY AND ADAM PARUSINSKI

On the Milnor classes of complex hypersurfaces 161LAURENTIU MAXIM

An introduction to intersection homology with general perversity functions 177GREG FRIEDMAN

The signature of singular spaces and its refinements to generalized homologytheories 223

MARKUS BANAGL

Intersection homology Wang sequence 251FILIPP LEVIKOV

An exponential history of functions with logarithmic growth 281MATT KERR AND GREGORY PEARLSTEIN

Motivic characteristic classes 375SHOJI YOKURA

Characteristic classes of mixed Hodge modules 419JORG SCHURMANN

Open problems 471

vii


Preface

This volume is based on lectures given at the Workshop on the Topology ofStratified Spaces that was held at the Mathematical SciencesResearch Institutein Berkeley, CA, from September 8 to September 12, 2008. Stratified spacesare usually not quite manifolds — they may possess singularities — but they arecomposed of manifold layers, the strata. Examples of such spaces include alge-braic varieties, quotients of manifolds and varieties by group actions, homotopystratified spaces, topological and piecewise linear pseudomanifolds, and evenmanifolds, augmented by filtrations that can arise, for example, via embeddingsand their singularities. In recent years, there has been extensive interest andsuccess in expanding to stratified spaces the triumphs of algebraic topology inmanifold theory, including the vast progress in the mid-twentieth century onsignatures, characteristic classes, surgery theories, and the special homologicalproperties of nonsingular analytic and algebraic varieties, such as the Kahlerpackage and Hodge theories. Such extensions from manifold theory to strati-fied space theory are rarely straightforward — they tend to involve the discoveryand study of subtle interactions between local and global behavior — but vastprogress has been made, particularly using such topological tools as intersectionhomology and the related analyticL2 cohomology. The goal of the workshopwas, and of this proceedings is, to provide an overview of this progress as well asof current research results, with a particular emphasis on communication acrossthe boundaries of the different fields of mathematics that encompass stratifiedspace research. Thus there is an emphasis in this volume on expository papersthat give introductions to and overviews of topics in the area of stratified spaces.

Four main areas were featured in the MSRI workshop:L2 cohomology and

Hodge theorems, topology of algebraic varieties, signature theory on singularspaces, and mixed Hodge theory and singularities. For the purpose of givingsome organization to the volume, we have grouped the papers roughly into thesetopics, although some papers overlap more than one area.

There are three papers on analysis and topology. The paper byDai is anintroduction toL

2 cohomology, which discusses some of the analytic consider-ations that arise in the study ofL

2 cohomology andL2 signatures on stratified

ix

x PREFACE

spaces as well as relationships between these analytic objects and topologicalanalogues. Building on these basics, the paper of Carron gives an example ofhow his idea of the exterior derivative on a space having “almost closed range”can be used to calculate theL2 cohomology in the case of two QALE-typespaces, the Hilbert schemes of two and of three points onCP

2. The paper byWaelder comes from a different angle. It considers “rigid” differential operatorson a smooth manifold commuting with anS1-action, and infinite dimensionalanalog of the Dirac operator, whose index yields the complexelliptic genus. Inthis survey, Waelder discusses how rigidity theorems for such operators are re-lated to the problem of defining Chern numbers on singular varieties. The paperof Waelder also fits under the second category: complex algebraic varieties.

The next section, on algebraic varieties, starts with the survey by Kovacs andSchwede, which gives an introduction to the study of singularities that are un-avoidable in classification problems of smooth algebraic varieties, and especiallyin the minimal model program, including log-canonical and Du Bois singular-ities. The paper of Libgober is an overview of the work of the author and hiscollaborators on extensions of the elliptic genus to singular varieties. It containsa new description of the class of holomorphic functions (quasi-Jacobi forms)that are elliptic genera of complex manifolds (possibly without the Calabi–Yaucondition) and includes a survey of recent developments such as the “higherelliptic genera”, which contain information on non-simplyconnected spaces.It also constructs the elliptic genus for singular real algebraic spaces. In theircontribution, McCrory and Parusinski make a thorough study of the weight fil-tration on the Borel–Moore cohomology of real algebraic varieties. The authorsgive several descriptions of this filtration and show some nice applications oftheir construction to real algebraic and analytic geometry. Finally, the paper ofMaxim gives a new formulation of results about Milnor classes, which general-ize Parusinski’s Milnor numbers to non-isolated hypersurface singularities. Heconsiders these classes for singular hypersurfaces in complex manifolds and,using his new formulation, gives comparisons to the topologically definedL-classes of Cappell–Shaneson and Goresky–MacPherson.

Three papers discuss aspects of intersection homology and signatures on strat-ified spaces. The paper by Friedman is an expository survey ofperversities inintersection homology, starting with the classical perversities of Goresky andMacPherson and describing various ways in which these have been generalizedin the past three decades. It can also be read as a general introduction to in-tersection homology. Banagl’s contribution is an overviewof bordism invariantsignatures on singular spaces, including the category of Witt space, where thesignature on intersection cohomology provides a Witt-bordism invariant sig-nature, and Banagl’s new signature on a category of non-Wittspaces under a

PREFACE xi

new type of bordism. Levikov’s paper describes the existence of a Wang-typesequence for intersection homology, which has implications for the non-Wittsignatures defined by Banagl.

The section on mixed Hodge structures includes a comprehensive survey byKerr and Pearlstein of recent developments in the theory of normal functions.It begins with the classical theory of Griffiths, Steenbrink, and Zucker, and in-cludes the work of Green, Griffiths, and Kerr on limits of Abel–Jacobi mappings,the equivalence (due to Brosnan, Fang, Nie, and Pearlstein)of the Hodge con-jecture to a question about singularities of certain normalfunctions; work byBrosnan and Pearlstein on the algebraicity of the zero locusof an admissiblenormal function; and the construction of a Neron model by Green, Griffiths,and Kerr, and by Brosnan, Pearlstein, and Saito. This surveyincludes severalconcrete examples and an extensive bibliography, and it concludes with a dis-cussion of open questions in the field. The papers by Yokura and Schurmann,respectively, describe recent work by the authors, together with Brasselet, whichgives a positive answer to the question of MacPherson as to whether there existsa unified theory of characteristic classes for singular varieties that is analogousto the classical Hirzebruch theory. Yokura’s paper emphasizes the motivic touchof the story, whereas Schurmann’s approach relies on Saito’s powerful theoryof algebraic mixed Hodge modules.

This volume concludes with an annotated list of problems proposed by par-ticipants in the workshop.

Neither the workshop nor this volume could have come about without thehelp of many people. Firstly, MSRI provided funding for the workshop and itsstaff helped enormously in making it a success. Secondly, Silvio Levy helpedarrange publication of this volume and gave us good advice onhow to makeit as useful a book as possible. Finally, several of our colleagues have beeninsightful and efficient referees for the various papers herein. We are grateful toeveryone for their assistance. We hope that readers will findthis volume usefuland inspiring.

Greg Friedman, Eugenie Hunsicker,Anatoly Libgober, Laurentiu Maxim

September 2009


An introduction to L2 cohomologyXIANZHE DAI

ABSTRACT. After a quick introduction toL2 cohomology, we discuss recentjoint work with Jeff Cheeger where we study, from a mostly topological stand-point, theL2-signature of certain spaces with nonisolated conical singularities.The contribution from the singularities is identified with atopological invariantof the link fibration of the singularities, involving the spectral sequence of thelink fibration.

This paper consists of two parts. In the first, we give an introduction toL2 cohomology. This is partly based on [8]. We focus on the analytic aspectof L2 cohomology theory. For the topological story, we refer to [1; 22; 31]and of course the original papers [16; 17]. For the history and comprehensiveliterature, see [29]. The second part is based on our joint work with Jeff Cheeger[11], which gives the contribution to theL2 signature from nonisolated conicalsingularity.

It is a pleasure to thank Eugenie Hunsicker for numerous comments and sug-gestions.

1. L2 cohomology: what and why

What is L2 cohomology? The de Rham theorem provides one of the most use-ful connections between the topological and differential structure of a manifold.The differential structure enters the de Rham complex, which is the cochaincomplex of smooth exterior differential forms on a manifoldM , with the exte-rior derivative as the differential:

0!˝0.M /d! ˝1.M /

d! ˝2.M /

d! ˝3.M /! � � �

Partially supported by NSF and CNSF..

1

2 XIANZHE DAI

The de Rham Theorem says that the de Rham cohomology, the cohomology

of the de Rham complex,H kdR.M /

defD kerdk=Im dk�1, is isomorphic to the

singular cohomology:H k

dR.M /ŠH k.M IR/:

The situation can be further rigidified by introducing geometry into the pic-ture. Letg be a Riemannian metric onM . Theng induces anL2-metric on˝k.M /. As usual, letı denote the formal adjoint ofd . In terms of a choice oflocal orientation forM , we haveıD˙�d�, where� is the Hodge star operator.Define the Hodge Laplacian to be

�D dıC ıd:

A differential form! is harmonic if�! D 0.The great theorem of Hodge then states that, for a closed Riemannian mani-

fold M , every de Rham cohomology class is represented by a unique harmonicform. This theorem provides a direct bridge between topology and analysis ofmanifolds through geometry, and has found many remarkable applications.

Naturally, then, one would like to extend the theory to noncompact manifoldsand manifolds with singularity. The de Rham cohomology is still defined (onewould restrict to the smooth open submanifold of a manifold with singularity).However, it does not capture the information at infinity or atthe singularity.

One way of remedying this is to restrict to a subcomplex of theusual de Rhamcomplex, namely that of the square integrable differentialforms — this leads usto L2 cohomology.

More precisely, let.Y;g/ denote an open (possibly incomplete) Riemannianmanifold, let˝i D˝i.Y / be the space ofC 1 i-forms onY andL2 DL2.Y /

the L2 completion of˝i with respect to theL2-metric. Defined to be theexterior differential with the domain

domd D f˛ 2˝i.Y /\L2.Y /I d˛ 2L2.Y /g:

Put˝i.2/.Y /D˝

i.Y /\L2.Y /:

Then one has the cochain complex

0!˝0.2/.Y /

d! ˝1

.2/.Y /d! ˝2

.2/.Y /d! ˝3

.2/.Y /! � � � :

The L2-cohomology ofY is defined to be the cohomology of this cochaincomplex:

H i.2/.Y /D kerdi=Im di�1 :

Thus defined, theL2 cohomology is in general no longer a topological invariant.However, theL2 cohomology depends only on the quasi-isometry class of themetric.

AN INTRODUCTION TO L2 COHOMOLOGY 3

EXAMPLES. � The real line: For the real lineR with the standard metric,

H i.2/.R/D 0 if i D 0;

H i.2/.R/ is infinite-dimensional ifi D 1:

For the first part, this is because constant functions can never beL2, unlessthey are zero. For the second part, a1-form �.x/ dx, with �.x/ havingcompact support, is obviously closed andL2, but can never be the exteriorderivative of anL2 function, unless the total integral of� is zero.

� Finite cone: LetC.N / D CŒ0;1�.N / D .0; 1/ � N , whereN is a closedmanifold of dimensionn, with the conical metricg D dr2C r2gN . Then aresult of Cheeger [8] gives

H i.2/.C.N //D

�

H i.N / if i < .nC 1/=2;

0 if i � .nC 1/=2:

Intuitively this can be explained by the fact that some of thedifferentialforms that define classes for the cylinderN �.0; 1/ cannot beL2 on the coneif their degrees are too big. More specifically, let! be ani-form onN andextend it trivially toC.N /, so! is constant along the radial direction. Then

Z

C.N /

j!j2gd volg DZ 1

0

Z

N

j!jgNrn�2idx dr:

Thus, the integral is infinite ifi � .nC 1/=2.

As we mentioned, theL2 cohomology is in general no longer a topologicalinvariant. Now clearly, there is a natural map

H i.2/.Y /�H i.Y;R/

via the usual de Rham cohomology. However, this map is generally neitherinjective nor surjective. On the other hand, in the case when.Y;g/ is a compactRiemannian manifold with corner (for a precise definition see the article byGilles Carron in this volume), the map above is an isomorphism because theL2

condition is automatically satisfied for any smooth forms.Also, another natural map is from the compact supported cohomology to the

L2 cohomology:H i

c .Y /�H i.2/.Y /:

As above, this map is also neither injective nor surjective in general.Instead, theL2 cohomology of singular spaces is intimately related to the in-

tersection cohomology of Goresky–MacPherson ( [16; 17]; see also Greg Fried-man’s article in this volume for the intersection cohomology). This connectionwas pointed out by Dennis Sullivan, who observed that Cheeger’s local com-putation ofL2 cohomology for isolated conical singularity agrees with that of

4 XIANZHE DAI

Goresky–MacPherson for the middle intersection homology.In [8], Cheegerestablished the isomorphism of the two cohomology theoriesfor admissiblepseudomanifolds. One of the fundamental questions has beenthe topologicalinterpretation of theL2 cohomology in terms of the intersection cohomology ofGoresky–MacPherson.

Reduced L2 cohomology and L2 harmonic forms. In analysis, one usuallyworks with complete spaces. That means, in our case, the fullL2 space insteadof just smooth forms which areL2. Now the coboundary operatord has welldefined strong closured in L2: ˛ 2 domd andd˛ D � if there is a sequence

j 2 domd such that j ! ˛ andd j ! � in L2. (The strong closure is tomaked a closed operator. There are other notions of closures and extensions,as in [15] for instance.) Similarly,ı has the strong closureNı.

One can also define theL2-cohomology using the strong closured . Thus,define

H i.2/;#.Y /D kerd i=Im d i�1 :

Then the natural map

�.2/ W H i.2/.Y /�H i

.2/;#.Y /

turns out to be always an isomorphism [8].This is good, but does not produce any new information . . . yet! The crucial

observation is that, in general, the image ofd need not be closed. This leads tothe notion of reducedL2-cohomology, which is defined by quotienting out bythe closure instead:

NH i.2/.Y /D kerd i=Im d i�1 :

The reducedL2-cohomology is generally not a cohomology theory but it isintimately related to Hodge theory, as we will see.

Now we define the space ofL2-harmonici-formsHi.2/.Y / to be

Hi.2/.Y /D f� 2˝

i \L2I d� D ı� D 0g:

Some authors define theL2-harmonic forms differently; compare [31]. The def-initions coincide when the manifold is complete. The advantage of our definitionis that, whenY is oriented, the Hodge star operator induces

� WHi.2/.Y /!H

n�i.2/ .Y /;

which is naturally the Poincare duality isomorphism.Now the big question is: Do we still have a Hodge theorem?


Kodaira decomposition, L2 Stokes and Hodge theorems. To answer thequestion, let’s look at the natural map, the Hodge map

Hi.2/.Y /�H i

.2/.Y /:

The question becomes: When is this map an isomorphism? Following Cheeger[8], when the Hodge map is an isomorphism, we will say that thestrong Hodgetheorem holds.

The most basic result in this direction is the Kodaira decomposition [23] (seealso [14]),

L2 DHi.2/˚ d�i�1

0˚ ı�iC1

0;

an orthogonal decomposition which leaves invariant the subspaces of smoothforms. Here subscript0 denotes having compact support. This result is essen-tially the elliptic regularity.

It follows from the Kodaira decomposition that

kerd i DHi.2/˚ d�i�1

0:

Therefore the question reduces to what the space Imd i�1 is in the decomposi-tion. We divide the discussion into two parts:

SURJECTIVITY. If Im d is closed, then Imd � d�i�10

. Hence, the Hodge mapis surjective in this case.

In particular, this holds if theL2-cohomology is finite-dimensional.

INJECTIVITY. The issue of injectivity of the Hodge map has to do with theL2

Stokes theorem. We say that Stokes’ theorem holds forY in theL2 sense if

hd˛; ˇi D h˛; Nıˇi

for all ˛ 2 domd , ˇ 2 dom Nı; or equivalently, if

hd˛; ˇi D h˛; ıˇi

˛ 2 domd , ˇ 2 domı.If the L2 Stokes theorem holds, one has

Hi.2/.Y /? Im d i�1;

so the Hodge map is injective in this case. Moreover,

H i.2/.Y /DH

i.2/.Y /˚ Im d i�1=Im d i�1:

Here, by the closed graph theorem, the last summand is either0 or infinite-dimensional. Note also, since it follows that

Hi.2/.Y /? Im d i�1 ;

6 XIANZHE DAI

thatNH i.2/.Y /ŠH

i.2/.Y /:

That is, when theL2 Stokes theorem holds, the reducedL2 cohomology issimply the space ofL2 harmonic forms.

Summarizing, if theL2-cohomology ofY has finite dimension and Stokes’theorem holds onY in theL2-sense, then the Hodge theorem holds in this case,and theL2-cohomology ofY is isomorphic to the space ofL2-harmonic forms.Therefore, whenY is orientable, Poincare duality holds as well. Consequently,theL2 signature ofY is well defined in this case.

There are several now classical results regarding theL2 Stokes theorem.Gaffney [15] showed that theL2 Stokes theorem holds for complete Riemann-ian manifolds. On the other hand, for manifolds with conicalsingularityM D

M0 [ C.N /, the general result of Cheeger [9] says that theL2 Stokes theo-rem holds provided thatL2 Stokes holds forN and in addition the middle-dimensional (L2) cohomology group ofN vanishes if dimN is even. In partic-ular, if N is a closed manifold of odd dimension, orH dimN=2.N /D 0 if dim N

is even, theL2 Stokes theorem holds forM .

REMARK . There are various extensions ofL2, for cohomology example, co-homology with coefficients or Dolbeault cohomology for complex manifolds.

2. L2 signature of nonisolated conical singularities

Nonisolated conical singularities.We now consider manifolds with nonisolatedconical singularity whose strata are themselves smooth manifolds. In otherwords, singularities are of the following type:

(i) The singular stratum consists of disjoint unions of smooth submanifolds.(ii) The singularity structure along the normal directionsis conical.

More precisely, a neighborhood of a singular stratum of positive dimension canbe described as follows. Let

Zn!M m �! Bl (2-1)

be a fibration of closed oriented smooth manifolds. Denote byC�M the map-ping cylinder of the map� W M !B. This is obtained from the given fibrationby attaching a cone to each of the fibers. Indeed, we have

CŒ0;1�.Z/! C�M !B :

The spaceC�M also comes with a natural quasi-isometry class of metrics.A metric can be obtained by choosing a submersion metric onM :

gM D ��gBCgZ :


Then, on the nonsingular part ofC�M , we take the metric

g1 D dr2C��gBC r2gZ : (2-2)

The general class of spaces with nonisolated conical singularities as abovecan be described as follows. A spaceX in the class will be of the form

X DX0[X1[ � � � [Xk ;

whereX0 is a compact smooth manifold with boundary, and eachXi (for i D

1; : : : ; k) is the associated mapping cylinderC�iMi for some fibration.Mi ; �i/,

as above.More generally, one can consider the iterated constructionwhere we allow

manifolds in our initial fibration to have singularities of the type consideredabove. However, we will restrict ourselves to the simplest situation where theinitial fibrations are all modeled on smooth manifolds.

REMARK . An n-dimensional stratified pseudomanifoldX is a topological spacetogether with a filtration by closed subspaces

X DXn DXn�1 �Xn�2 � � � � �X1 �X0

such that for each pointp 2 Xi �Xi�1 there is a distinguished neighborhoodU in X which is filtered homeomorphic toC.L/�Bi for a compact stratifiedpseudomanifoldL of dimensionn�i�1. Thei-dimensional stratumXi�Xi�1

is ani-dimensional manifold. Aconical metriconX is a Riemannian metric onthe regular set ofX such that on each distinguished neighborhood it is quasi-isometric to a metric of the type (2-2) withBDBi ;ZDL andgB the standardmetric onBi , gZ a conical metric onL. Such conical metrics always exist ona stratified pseudomanifold.

L2 signature of generalized Thom spaces. A generalized Thom spaceT isobtained by coning off the boundary of the spaceC�M .

Namely,

T D C�M [M C.M /

is a compact stratified pseudomanifold with two singular strata:B and a singlepoint (unlessB is a sphere).

EXAMPLE . Let ��! B be a vector bundle of rankk. We have the associated

sphere bundle

Sk�1! S.�/�! B:

The generalized Thom space constructed out of this fibrationcoincides with theusual Thom space equipped with a natural metric.

8 XIANZHE DAI

Now consider the generalized Thom space constructed from anoriented fibration(2-1) of closed manifolds, i.e., both the baseB and fiberZ are closed orientedmanifolds and so is the total spaceM . ThenT will be a compact orientedstratified pseudomanifold with two singular strata. Since we are interested intheL2 signature, we assume that the dimension ofM is odd (so dimT is even).In addition, we assume the Witt conditions; namely, either the dimension of thefibers is odd or its middle-dimensional cohomology vanishes. Under the Wittconditions, the strong Hodge theorem holds forT . Hence theL2 signature ofT is well defined.

QUESTION. What is theL2 signature ofT ?

Let’s go back to the case of the usual Thom space.

EXAMPLE (continued). In this case,

sign.2/.T /D�sign.D.�//;

the signature of the disk bundleD.�/ (as a manifold with boundary).Let˚ denote the Thom class and� the Euler class. Then the Thom isomor-

phism gives the commutative diagram

H �Ck.D.�/;S.�// ˝ H �Ck.D.�/;S.�// - R

H �.B/

��. � /[˚

6

˝ H �.B/

��. � /[˚

6

- R

� - Œ� [ [��ŒB� :

Thus, sign.2/.T / is the signature of this bilinear form onH �.B/.

We now introduce the topological invariant which gives theL2-signature for ageneralized Thom space. In [13], in studying adiabatic limits of eta invariants,we introduced a global topological invariant associated with a fibration. (Foradiabatic limits of eta invariants, see also [32; 5; 10; 3].)Let .Er ; dr / be theEr -term with differential,dr , of the Leray spectral sequence of the fibration(2-1) in the construction of the generalized Thom spaceT . Define a pairing

Er ˝Er � R

�˝ ’h� � dr ; �r i;

where�r is a basis forEmr naturally constructed from the orientation. In case

mD 4k�1, when restricted toE.m�1/=2r , this pairing becomes symmetric. We


define�r to be the signature of this symmetric pairing and put

� DX

r�2

�r :

When the fibration is a sphere bundle with the typical fiber a.k � 1/-dimen-sional sphere, the spectral sequence satisfiesE2D � � � DEk , EkC1DE1 withd2 D � � � D dk�1 D 0; dk. / D [ �. Hence� coincides with the signatureof the bilinear form from the Thom isomorphism theorem. The main result of[11] is this:

THEOREM 1 (CHEEGER–DAI ). Assume either the fiberZ is odd-dimensionalor its middle-dimensional cohomology vanishes. Then theL2-signature of thegeneralized Thom spaceT is equal to�� :

sign.2/.T /D��:

In spirit, our proof of the theorem follows the example of thesphere bundleof a vector bundle. Thus, we first establish an analog of Thom’s isomorphismtheorem in the context of generalized Thom spaces. In part, this consists ofidentifying theL2-cohomology ofT in terms of the spectral sequence of theoriginal fibration; see [11] for complete details.

COROLLARY 2. For a compact oriented spaceX with nonisolated conical sin-gularity satisfying the Witt conditions, theL2-signature is given by

sign.2/.X /D sign.X0/C

kX

iD1

�.Xi/ :

The study of theL2-cohomology of the type of spaces with conical singularitiesdiscussed here turns out to be related to work on theL2-cohomology of non-compact hyperkahler manifolds which is motivated by Sen’s conjecture; see, forexample, [19] and [18]. Hyperkahler manifolds often arise as moduli spaces of(gravitational) instantons and monopoles, and so-called S-duality predicts thedimension of theL2-cohomology of these moduli spaces (Sen’s conjecture).Many of these spaces can be compactified to give a space with nonisolatedconical singularities. In such cases, our results can be applied. We also referthe reader to the work [18] of Hausel, Hunsicker and Mazzeo, which studiestheL2-cohomology andL2-harmonic forms of noncompact spaces with fiberedgeometric ends and their relation to the intersection cohomology of the compact-ification. Various applications related to Sen’s conjecture are also consideredthere.

10 XIANZHE DAI

Combining the index theorem of [4] with our topological computation of theL2-signature ofT , we recover the following adiabatic limit formula of [13]; seealso [32; 5; 9; 3].

COROLLARY 3. Assume that the fiberZ is odd-dimensional. Then we have thefollowing adiabatic limit formula for the eta invariant of the signature operator:

lim"!0

�.AM;"/D

Z

B

L

�

RB

2�

�

^ Q�C �:

In the general case, that is, with no dimension restriction on the fiber, theL2-signature for generalized Thom spaces is discussed in [21]. In particu-lar, Theorem 1 is proved for the general case in [21]. However, one of theingredients there is the adiabatic limit formula of [13], rather than the directtopological approach taken here. One of our original motivations was to givea simple topological proof of the adiabatic limit formula. In [20], the methodsand techniques in [11] are used in the more general situationto derive a veryinteresting topological interpretation for the invariant�r . On the other hand, in[7], our result on the generalized Thom space, together withthe result in [13],is used to derive the signature formula for manifolds with nonisolated conicalsingularity.

References

[1] A. Borel et al.Intersection cohomology. Notes on the seminar held at the Universityof Bern, Bern, 1983. Reprint of the 1984 edition. Birkhauser, Boston, 2008.

[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and riemannian ge-ometry, I–III. Math. Proc. Cambridge Philos. Soc.77(1975):43–69, 78(1975):405–432, 79(1976):71–99.

[3] J.-M. Bismut and J. Cheeger.�-invariants and their adiabatic limits.Jour. Amer.Math. Soc.2:33–70, 1989.

[4] J.-M. Bismut and J. Cheeger. Remarks on families index theorem for manifolds withboundary.Differential geometry, 59–83, Pitman Monogr. Surveys Pure Appl. Math.,52, Longman Sci. Tech., Harlow, 1991 eds. Blaine Lawson and Keti Tenenbaum.

[5] J.-M. Bismut and D. S. Freed. The analysis of elliptic families I; II. Commun. Math.Phys.106:159–167, 107:103–163, 1986.

[6] R. Bott and L. Tu.Differential forms in algebraic topology. Graduate Texts inMathematics, 82. Springer, New York-Berlin, 1982.

[7] J. Bruning. The signature operator on manifolds with a conical singular stratum.Asterisque, to appear.

[8] J. Cheeger. On the Hodge theory of Riemannian pseudomanifolds.Geometry of theLaplace operator, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I.36:91–146, 1980.


[9] J. Cheeger. Spectral geometry of singular Riemannian spaces.J. Diff. Geom.18:575–657, 1983.

[10] J. Cheeger. Eta invariants, the adiabatic approximation and conical singularities.J.Diff. Geom.26:175–211, 1987.

[11] J. Cheeger and X. Dai.L2-cohomology of spaces of non-isolated conical sin-gularity and non-multiplicativity of signature,Riemannian topology and geometricstructures on manifolds, 1–24. Progress in Mathematics, 271, Birkhauser, Boston,2009,

[12] J. Cheeger, M. Goresky, and R. MacPherson.L2-cohomology and intersectionhomology of singular algebraic varieties.Seminar on Differential Geometry, Ann. ofMath. Stud., 102, Princeton Univ. Press, Princeton, N.J. pages 303–340, 1982.

[13] X. Dai. Adiabatic limits, nonmultiplicativity of signature, and Leray spectralsequence.J. Amer. Math. Soc.4:265–321, 1991.

[14] G. de Rham.Differentiable manifolds: forms, currents, harmonic forms. Grund-lehren der Mathematischen Wissenschaften, 266. Springer,Berlin, 1984.

[15] M. Gaffney. A special Stokes’s theorem for complete Riemannian manifolds.Ann.of Math.(2) 60, (1954). 140–145.

[16] M. Goresky and R. MacPherson. Intersection homology theory.Topology, 19:135–162, 1980.

[17] M. Goresky and R. MacPherson. Intersection homology II. Invent. Math.71:77–129, 1983.

[18] T. Hausel, E. Hunsicker, and R. Mazzeo. The Hodge cohomology of gravitationalinstantons.Duke Math. J.122(3):485–548, 2004.

[19] N. Hitchin. L2-cohomology of hyperkahler quotients.Comm. Math. Phys.211(1):153–165, 2000.

[20] E. Hunsicker. Hodge and signature theorems for a familyof manifolds withfibration boundary.Geometry and Topology, 11:1581–1622, 2007.

[21] E. Hunsicker, R. Mazzeo. Harmonic forms on manifolds with edges.Int. Math.Res. Not.2005, no. 52, 3229–3272.

[22] F. Kirwan. An introduction to intersection homology theory. Pitman ResearchNotes in Mathematics, 187. Longman, Harlow.

[23] K. Kodaira. Harmonic fields in Riemannian manifolds (generalized potentialtheory).Ann. of Math.(2) 50, (1949). 587–665.

[24] E. Looijenga.L2-cohomology of locally symmetric varieties.Compositio Math.67:3–20, 1988.

[25] R. Mazzeo. The Hodge cohomology of a conformally compact metric. J. Diff.Geom.28:309–339, 1988.

[26] R. Mazzeo and R. Melrose. The adiabatic limit, Hodge cohomology and Leray’sspectral sequence for a fibration.J. Diff. Geom.31(1):185–213, 1990.

12 XIANZHE DAI

[27] R. Mazzeo and R. Phillips. Hodge theory on hyperbolic manifolds.Duke Math. J.60:509–559, 1990.

[28] W. Pardon and M. Stern.L2-N@-cohomology of complex projective varieties.J.Amer. Math. Soc.4(3):603–621, 1991.

[29] S. Kleiman. The development of intersection homology theory.Pure Appl. Math.Q. 3(1/3):225–282, 2007.

[30] L. Saper and M. Stern.L2-cohomology of arithmetic varieties.Ann. of Math.132(2):1–69, 1990.

[31] L. Saper and S. Zucker. An introduction toL2-cohomology.Several complexvariables and complex geometry, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math.Soc., Providence, RI, 519–534, 1991.

[32] E. Witten. Global gravitational anomalies.Comm. Math. Phys.100:197–229,1985.

[33] S. Zucker. Hodge theory with degenerating coefficients. L2 cohomology in thePoincare metric.Ann. of Math.(2) 109:415–476, 1979.

[34] S. Zucker.L2 cohomology of warped products and arithmetic groups.Invent.Math.70:169–218, 1982.

X IANZHE DAI

MATHEMATICS DEPARTMENT

UNIVERSITY OF CALIFORNIA , SANTA BARBARA

SANTA BARBARA , CA 93106UNITED STATES

[email protected]


The almost closed range conditionGILLES CARRON

ABSTRACT. The almost closed range condition is presented and we explainhow this notion can be used to give a topological interpretation of the space ofL2 harmonic forms on the Hilbert schemes of2 and3 points onC

2.

A Jacques

1. Introduction

When .M;g/ is a compact manifold the celebrated theorem of Hodge andde Rham says that the spaces ofL2 harmonic forms onM are isomorphic tothe cohomology spaces ofM ; that is, if we denote by

Hk.M;g/D

˚

˛ 2L2g.�

kM /; d˛ D d�˛ D 0

the space ofL2 harmonick-forms,1 then we have a natural isomorphism

Hk.M;g/'H k.M;R/:

When.M;g/ is noncompact but complete, the spaces ofL2 harmonic formshave an interpretation in terms of reducedL2 cohomology. A general and naivequestion is to understand how we can give some topological interpretation forthese spaces ofL2 harmonic forms. There are many results, as well as pre-dictions and conjectures, in this direction. For instance,Zucker’s conjecture[32] about locally symmetric Hermitian spaces, eventuallysolved by E. Looi-jenga, L. Saper and M. Stern [18; 27] and extended by A. Nair [22], and therecent result of L. Saper [25; 26], as well as results for manifolds with flat ends[6], manifolds with cylindrical end [2], and negatively curved manifolds withfinite volume [17; 30; 31]. Also,L2 harmonic forms have some significance

1Hered� is the formal adjoint of the exterior differentiation operator d for theL2 structure induced bythe metricg.

13

14 GILLES CARRON

in modern physics and there are several predictions based ona duality arisingin string theory: for instance there is Sen’s conjecture about the moduli spaceof magnetic monopoles [28] and the Vafa–Witten conjecture about Nakajima’squiver manifolds [29; 13].

WhenM has a locally finite open coveringM DS

˛ U˛ admitting a par-tition of unity with bounded gradient such that on any ofM , U˛, U˛ \ Uˇ,U˛\Uˇ\U , . . . theL2-range ofd is closed, then we can sometimes use sheafcohomology to obtain a topological interpretation of the space ofL2 harmonicforms. However, this is not always possible. Several tools have been developedin order to circumvent this difficulty. For instance, pseudodifferential calculihave been used successfully in several situations [19; 20; 21; 14]. Here wepresent the notion of almost closed range ford which was introduced in [7].We give some general results (including a Mayer–Vietoris sequence) that aretrue if this almost closed range condition is satisfied. We also explain that thiscondition has to be used with some care. In order to illustrate how this notionis used in [7], we explain the arguments (and amongst them thealmost closedrange condition) leading to the topological interpretation of the space ofL2

harmonic forms on the Hilbert schemes of2 and3 points onC2.

2. L2 cohomology

We start with basic definitions, to present the setting and fixnotation.

2.1. Definitions. Let .M n;g/ be an oriented Riemannian manifold. We endowit with a smooth positive measure� dvolg (where� is a positive smooth func-tion), so that we can define the spaceL2

�.�kM / of differentialk-forms which

are inL2�.�

kM /. This is a Hilbert space when endowed with the norm

k˛k2� WD

Z

M

j˛.x/j2g � dvolg.x/:

The associated Hermitian scalar product will be denoted byh : ; : i�.We introduce the spaceZk

�.M / of L2� k-forms that are weakly closed:

Zk�.M / WD

˚

˛ 2L2�.�

kM / WR

M ˛^ d' D 0 for all ' 2 C 1

0.�n�1�kM /

The spaceZk�.M / is in fact a subspace of the (maximal) domain ofd ,

Zk�.M /�D

k�.d/;

whereDk�.d/ is the maximal domain ofd on L2

�.�kM /. This is the space of

˛ 2L2�.�

kM / such that there is a constantC withˇ

ˇ

ˇ

ˇ

Z

M

˛^ d'

ˇ

ˇ

ˇ

ˇ

� Ck'k� for all ' 2 C 1

0 .�n�1�kM /:

THE ALMOST CLOSED RANGE CONDITION 15

When˛ 2Dk�.d/ we can defined˛ 2L2

�.�kC1M / by duality:

Z

M

d˛^' D .�1/kC1

Z

M

˛^ d' for all ' 2 C 1

0 .�n�1�kM /:

By definition we havedDk�1� .d/�Zk

�.M /, butdDk�1� .d/ is not necessarily a

closed subspace ofL2�.�

kM /. If we introduce the spaceBk�.M /DdDk�1

� .d/,thenBk

�.M /�Zk�.M / sinceZk

�.M / is a closed subspace ofL2�.�

kM /.

DEFINITION 2.1. Thek-th reducedL2� cohomology space is the quotient

Hk�.M /D

Zk�.M /

Bk�.M /

:

Thek-th L2� cohomology space is the quotient

Zk�.M /

dDk�1� .d/

;

which is not a Hilbert space but satisfies other good properties of a cohomologytheory; for instance, a Mayer–Vietoris sequence holds forL2

� cohomology.Our aim is to circumvent the fact that in general we have problems computing

reducedL2� from local calculations because the Mayer–Vietoris exact sequence

does not hold in the reduced setting.

2.2. Some general properties of reducedL2� cohomology

Quasi-isometry invariance.The first general fact is a consequence of the defini-tion: L2

� (reduced or not) cohomology spaces depend only on theL2� topology;

hence ifg0 andg1 are two Riemannian metrics such that"g0� g1� g0=" for acertain" > 0, and if�0; �1 are positive smooth functions such that�0=�1 and�1=�0 are bounded, then

Hk�0.M;g0/DH

k�1.M;g1/:

Smooth forms inL2 cohomology.Using de Rham’s smoothing operator of (see[10] and also [9]), we can show that reduced and nonreducedL2

� cohomologycan be computed using only smooth forms; that is,

Hk�.M /'

Zk�.M /\C 1.�kM /

dDk�1� .d/\C 1.�kM /\C 1.�kM /

:

This smoothing argument also shows that ifM is a closed manifold, reducedand nonreducedL2

� cohomology are both isomorphic to de Rham cohomology.The smoothing operator gives additional results in the following setting: As-

sume thatM is an open subset in a manifoldN such that near every point of the

16 GILLES CARRON

boundaryp 2 @M DM nM there is a submersionxD .x1; : : : ;xk/ WU !Rk

on a neighborhood ofp such thatx.p/D 0 andU \M Dfx1> 0; : : : ;xk > 0g.Such a manifold will be called a manifold with corners. Consider a Riemannianmetric onM which extends smoothly toN . Moreover, assume that.M ;g/ ismetrically complete, that is, for anyo2M andr>0 then the closureB.o; r/\M

is compact inM . This is automatically the case wheng extends to a smoothgeodesically complete metric onN .

Then we can define two spaces of smooth forms:C 1

0.�kM / is the set of

smooth forms with compact support inM andC 1

0.�kM / is the set of smooth

forms with compact support inM . This is illustrated in Figure 1. Then asmoothing argument shows:

PROPOSITION 2.2. If .M;g/ is a Riemannian manifold with corner whoseclosure is metrically complete, thenC 1

0.�k�1M / is dense inDk�1

� .d/ whenthe domain ofd is endowed with the graph norm:

˛‘

q

k˛k2�Ckd˛k2�:

2.3. Harmonic forms andL2� cohomology. When the Riemannian manifold

.M;g/ is geodesically complete(hence boundaryless), reducedL2� cohomology

has an interpretation in terms of appropriate harmonicL2� forms. We introduce

d��, the formalL2

� adjoint of the operatord ; it is defined through the integrationby parts formula

hd˛; ˇi�Dh˛; d�

�ˇi� for all ˛ 2 C 1

0 .�kM / andˇ 2 C 1

0 .�kC1M /;

��

��

��

��

Figure 1. The support of an element of C 1

0.�kM / and the support of an

element of C 1

0 .�kM /. Here M is the positive quadrant fx1 � 0;x2 � 0g

in the plane.


Then we have

Hk�.M /'

˚

˛ 2L2�.�

kM / W d˛ D d�

�˛ D 0

D˚

˛ 2L2�.�

kM / W .d�

�d C dd�

�/˛ D 0

:

3. The almost closed range condition

From now we assume that.M;g/ is a manifold with corner whose closure ismetrically complete.

3.1. Good primitives. A natural question, which leads to a better understandingof reducedL2

� cohomology, is how anL2� smooth closed form

˛ 2L2.�kM /\C 1.�kM /

can be zero in reducedL2� cohomology.

A result of de Rham [10, Theorem 24] implies that for such an˛, there isalways a 2 C 1.�kM / such that

˛ D dˇ;

but thisˇ will not generally be inL2. By Proposition 2.2, the vanishing of thereducedL2

� cohomology class of is equivalent to the existence of a sequenceof smooth forms j 2 C 1

0.�k�1M / such that

dˇ DL2�- lim

j!1

d j :

Hence the problem is to understand what growth conditions onthe primitiveˇ imply the existence of such a sequence of smooth compactly supported forms,. j /.

It is clear that ifˇ 2 L2� then the class of D dˇ is zero inHk

�.M /. Anatural way to obtain a more general condition is to find a sequence of cut-off2

Lipschitz functions�j satisfying the following conditions:

� �j tends to1 uniformly on the compact sets ofM .� L2

�� limj!1 d�j ^ˇ D 0.

We always have�jˇ 2Dk�1� .d/ andd.�j ^ˇ/D d�j ^ˇC�j dˇ; hence these

conditions would imply D dˇ D L2� � limj!1 d j . The notion of para-

bolic weights, which we’re about to introduce is used to describe the regulationrequired on the growth of a primitive at infinity needed for this idea to work.

2i.e., with compact support onM .

18 GILLES CARRON

3.2. Parabolic weights

DEFINITION 3.1. A positive functionw WM ! .0;C1/ is called aparabolicweightwhen there is a function W .0;C1/! .0;C1/ such that

� for a fixed pointo 2M andr.x/D d.o;x/ we havew � 1

2.r/, and

�

Z 1

1dr

.r/DC1.

LEMMA 3.2. Assume that 2Zk�.M / satisfies D dˇ, withˇ 2L2

w� for some

parabolic weightw. Then the reducedL2� class of is zero.

We will not give a proof of this result, but we note that the parabolic conditionmakes possible the choice of a good sequence of cutoff functions as describedin the last subsection. We will instead explain where this definition comes from.Parabolicity for a weighted Riemannian manifold.M;g; �/ has several equiva-lent definitions in terms of Brownian motion, capacity, and existence of positiveGreen functions [1; 12]. We will only give the following definition:

DEFINITION 3.3. The weighted Riemannian manifold.M;g; �/ is called para-bolic if there is a sequence of cut-off Lipschitz functions�j such that

� �j tends to1 uniformly on the compact set ofM , and� limj!1

R

M jd�j j2�d volg D 0.

Here is a well known criterion that implies parabolicity.

PROPOSITION3.4. Let o be a fixed point in the weighted Riemannian manifold.M;g; �/ and let

L.r/D

Z

@B.o;r/

� d�g and V .r/D

Z

B.o;r/

� dvolg :

IfZ

1

1

dr

L.r/DC1 or

Z

1

1

rdr

V .r/DC1;

then.M;g; �/ is parabolic.

By definition, the parabolicity of the weighted manifold.M;g; jˇj2�/ impliesthat the reducedL2

� class ofdˇ D ˛ is zero. Whenw is a parabolic weight,we define the spaceCk�1

w;� .M / to be the set of 2L2w�.�

k�1M / such that theweak differential of is in L2

�:

Ck�1w;� .M / WD fˇ 2L2

w�.�k�1M / W dˇ 2L2

�g:

Then the parabolicity ofw implies thatd W Ck�1w;� .M /! Bk

�.M / is a boundedoperator.


3.3. The almost closed range properties

DEFINITION 3.5. We say that theL2� range ofd is almost closed in degreek

with respect tow whenw is a parabolic weight and

dCk�1w;� .M /D Bk

�.M /:

In other words, theL2� range ofd is almost closed in degreek with respect tow

if and only if everyL2� closed forms which is zero in reducedL2

� cohomologyhas aL2

w� primitive.

An example.Let˙ be an.n�1/-dimensional compact manifold with boundary,endowed with a smooth Riemannian metrich that extends smoothly to@˙ . Thetruncated coneC1.˙/D .1;C1/�˙ endowed with the conical metric

.dr/2C r2h

is a manifold with corner whose closure is metrically complete. Then we have:

PROPOSITION3.6. Consider the weight�a.r; �/D r2a. TheL2�a

cohomologyof C1.˙/ is given by

Hk�a.C1.˙//D

�

f0g if k � n2C a,

H k.˙/ if k > n2C a.

Introduce the two(parabolic) weightsw D 1=r2 andw D 1=.r2 log2.r C 1//.

� If k 6D n=2C a then theL2�a

range ofd is almost closed in degreek withrespect tow.

� If k D n=2Ca then in general theL2�a

range ofd is almost closed in degreek with respect tow.

� If k D n=2Ca andHn

2Ca�1.˙/D f0g, theL2

�arange ofd is almost closed

in degreek with respect tow.

The good news: a Mayer–Vietoris exact sequence.The almost closed range isconvenient because it implies a short Mayer–Vietoris exactsequence for reducedL2 cohomology:

PROPOSITION3.7. Assume thatM D U [ V and thatU; V andU \ V aremanifolds with corners whose closures are metrically complete. Assume that fora parabolic weightw WM ! .0;C1/, theL2

� range ofd is almost closed indegreek with respect tow onM , U , V andU \V , and that the sequence

f0g ! Ck�1w;� .M /

r�

� Ck�1w;� .U /˚ C

k�1w;� .V /

ı� C

k�1w;� .U \V /! f0g

is exact. Then we have the short Mayer–Vietoris exact sequence

Hk�1w� .U /˚H

k�1w� .V /

ı�H

k�1w� .U \V /

b�H

k�.M /

r�

�Hk�.U /˚H

k�.V /

ı�H

k�.U \V /:

20 GILLES CARRON

We will not give the proof of this result. In fact, the argument is relativelystraightforward. We have only to follow the proof of the exactness of the Mayer–Vietoris sequence in de Rham cohomology in the compact case;the hypothesesmade here are the ones that are necessary to adapted these classical arguments.

The bad news.There is a difficulty with the assumption

Ck�1w;� .U /˚ C

k�1w;� .V /

ı� C

k�1w;� .U \V /! f0g:

For instance, letC1.˙/ be a truncated cone over a compact manifold withboundary.˙; h/. Now when˙ D zU [ zV , where zU ; zV ; zU \ zV are open withsmooth boundaries, then if we letU D C1. zU / andV D C1. zV /, we have thatfor the parabolic weightw D 1=r2, the sequence

Ck�1w;1 .U /˚ C

k�1w;1 .V /

ı� C

k�1w;1 .U \V /! f0g

is exact. However, for the parabolic weightwD1=.r2 log2.rC1//, the sequence

Ck�1w;1 .U /˚ C

k�1w;1 .V /

ı� C

k�1w;1 .U \V /! f0g

is not (necessarily) exact.From Proposition 3.6, we see that on a truncated cone we cannot always use

only the weightw. Thus we’ll have some difficulties using this exact sequence.

3.4. Comparison with other notions

With the nonparabolicity condition. In [4] and [5], we introduced the notionof nonparabolicity at infinity for the Dirac operator on a complete Riemannianmanifold and used it in [6] to compute theL2 cohomology of manifolds withflat ends. This condition is an extended Fredholmness condition. Specialized tothe case of the Gauss–Bonnet operator,dCd�

�, this condition is satisfied in thefollowing case:

PROPOSITION 3.8. Assume.M;g/ is a complete Riemannian manifold andthere is a weightw WM ! .0;C1/ and a compact setK �M such that

k˛kw� � k.d C d�

�/˛k� for all ˛ 2 C 1

0 .�k.M nK//;

Then the reducedL2� cohomology ofM is finite-dimensional. Moreover, for

any˛ 2 L2�.�

kM /, there existh 2 L2.�kM / such thatdh and d��h vanish,

ˇ 2L2w�.�

k�1M /, and 2L2w�.�

kC1M /, such that

˛ D hC dˇC d�

� :

Finally, for any˛ 2 Bk�.M /, there is a 2L2

w�.�k�1M / such that D dˇ.


Hence, under the assumptions of the Proposition 3.8, ifw is a parabolic weightthe L2

� range ofd is almost closed in degreek with respect tow. There is aclosely related result about the almost closed range condition [7]:

PROPOSITION 3.9. Assume that.M;g/ is a complete Riemannian manifoldand thatw WM ! .0;C1/ is a parabolic weight. Suppose there are a positiveconstantC and a compact setK �M such that

Ck˛k2w� � kd˛k2�Ckd

�

w�˛k2w� for all ˛ 2 C 1

0 .�k.M nK//: (3-1)

Then

� theL2� range ofd is almost closed in degreek with respect tow;

� theL2w� range ofd is almost closed in degreek � 1 with respect tow;

� the spaceH k�1w� .M / is finite-dimensional.

Moreover, these three properties imply the existence of a positive constantCand a compact setK �M such that the inequality(3-1)holds.

The first proposition is in fact a statement about the operator d C d��, whereas

the second is a statement aboutd .

With more classical cohomology theory.Let .X;g/ be a complete Riemannianmanifold, fix a degreek and assume that we have a sequence of weightswl

(which will depend onk in general) such thatwk D 1 and, for all degreesl , theL2wl�

range ofd is almost closed in degreel with respect towl�1=wl . Thenwe consider the complex

� � � ! Cl�1wl�1=wl ;wl�

.X /d! C

lwl =wlC1;wlC1�

.X /d! � � �

When the cohomology of this complex can be computed from a local computa-tion (that is, when there is a Poincare lemma characterizing the cohomology ofthis complex), then theL2

� cohomology ofX can be obtained from the degreek cohomology space of this complex. This method has been used successfullyby T. Hausel, E. Hunsicker and R. Mazzeo in [14] to obtain a topological inter-pretation of theL2

�D1cohomology of manifolds with fibered cusp ends or with

fibered boundary ends. However, in this case the proof is not simple and theauthors have to face the same kind of difficulty as the one we encountered onpage 20, essentially because the choice of primitive sometimes doesn’t lead toa complex whose cohomology follows from local computations. In this paper,the authors had to compare theL2

�D1cohomology with two other weighted

cohomologies,L2�, L2

1=�, with � D r " or � D e"r wherer is the function

given by distance to a fixed point; this comparison is made with an adaptedpseudodifferential calculus.

22 GILLES CARRON

4. The QALE geometry of the Hilbert scheme of 2 or 3 points

We will now describe the QALE geometry of the Hilbert scheme of 2 and3

points onC2. The Hilbert scheme ofn points onC

2, denoted by Hilbn0.C2/, is

a crepant resolution of the quotient of.C2/n0D˚

q 2 .C2/n;P

j qj D 0

by theaction of the symmetric groupSn, which acts by permutation of the indices:

� 2 Sn; q 2 .C2/n0; �:q D .q��1.1/; q��1.2/; : : : ; q��1.n//:

Hence we have a resolution of singularities map

� W Hilbn0.C

2/! .C2/n0=Sn:

4.1. The case of 2 points.For nD 2, we have

.C2/n0 D f.x;�x/ W x 2 C2gI

hence.C2/20=S2 ' C

2=f˙ Idg:

Now the crepant resolution ofC2=f˙ Idg is T �P1.C/; indeed, we have that the

cotangent bundle ofP1.C/ is the set of pairs.L; �/ whereL is a line inC2 and

� W C2! C2 a linear map such that the range of� is contained inL and such

that the kernel of� containsL. That is,� induces a linear map� W C2=L!L.In particular,T �P1.C/ nP1.C/ is identified with the set˚

� 2M2.C2/ W � 6D 0; � ı � D 0

D˚�

ac

b�a

�

W a2 D bc; .a; b; c/ 6D .0; 0; 0/

;

through the identification� ‘ .Im �; �/. This space is diffeomorphic to thequotient.C2 n f0g/=f˙ Idg through the map

˙.x;y/‘

�

xy y2

x2 �xy

�

:

T �P1.C/ carries a remarkable metric, the Eguchi–Hanson metric, which isKahler and Ricci flat [11; 3]. Moreover, this metric onT �P1.C/ n P1.C/ '

C2=f˙ Idg is asymptotic to the Euclidean metric. Such a metric is called asymp-

totically locally euclidean (ALE in short).

4.2. The case of 3 points.We can also understand the geometry of Hilb30.C

2/.Outside a compact set,.C2/3

0=S3 is a truncated cone overS7=S3 and the singular

set of the quotientS7=S3 pulls back toS7 as a disjoint union of 3 sub-spheresS3 given by the intersection ofS7 with the collision planes given by

Pi;j D f.q1; q2; q3/ 2 .C2/30 W qi D qj g; wherei < j:

This is illustrated in Figure 2, left. These three spheres are interchanged by theaction ofS3; hence the singular set ofS7=S3 is a sphereS3 and the geometry of


��

��

��

��

��

��

��

��

�

Figure 2. Left: The three collision planes in .C2/30. Right: S7=S3.

S7=S3 near the singular set is the one ofB2=f˙ Idg � S3, whereB2 is the unitball in C

2.Hence, as illustrated in Figure 2, right, outside a compact set, .C2/3

0=S3 is

the union of

� U , a truncated cone over.S7=S3/ nO, whereO is an"-neighborhood of thesingular set (hence homeomorphic toB2=f˙ Idg � S3),

�

˚

.x; v/ 2 C2=f˙ Idg �C

2 W jxj2Cjvj2 > 1; jxj< "jvj

.

The geometry at infinity of Hilb30.C2/ is the union of two open sets:

� U : a truncated cone over.S7=S3/ nO, and� V D

˚

.y; v/ 2 T �P1.C/� .C2 nB2/ W jyj< "jvj

, wherejyj is the pullbackto T �P1.C/ of the Euclidean distance;

that is, there is a compact setK�Hilb30.C

2/ such that Hilb30.C2/nKDU [V .

In [16], D. Joyce constructed a hyperkahler metricg on Hilb30.C

2/ (in partic-ular this is a Kahler and Ricci flat metric) which is quasi-asymptotically locallyeuclidean (QALE) asymptotic to.C2/3

0=S3, which means that

� on U , the truncated cone over.S7=S3/ nO, we have for alll 2 N

rl.g�eucl/DO� 1

r4Cl

�

;

wherer is the radial function on this cone.

� On V we have, for alll 2 N,

rl�

g� .gHilb2

0.C2/C eucl/

�

DO

�

1

jyj2Cl jvj2

�

:

24 GILLES CARRON

5. L2 cohomology of the Hilbert scheme of 2 or 3 points

We now explain how we can compute theL2 cohomology3 of the HilbertScheme of2 or 3 points with the almost closed range condition.

5.1. The case of 2 points.We use Proposition 3.6, which computed theL2

cohomology of truncated cones. Since Hilb20.C

2/D T �P1.C/ has real dimen-sion4, is oriented and has infinite volume, we easily get that theL2 cohomologyof Hilb2

0.C2/ is zero in degrees0 and4. Moreover, the Ricci curvature of the

Eguchi–Hansen metric is zero, so the Bochner formula implies that for anyL2

harmonic1-form ˛, we get

0D

Z

Hilb2

0.C2/

jd˛j2Cjd�˛j2 D

Z

Hilb2

0.C2/

jr˛j2:

Hence the space ofL2 harmonic1-forms is trivial and it remains only to computetheL2 cohomology of Hilb20.C

2/ in degree2. The main point is this:

LEMMA 5.1. The natural map from cohomology with compact support toL2

cohomology is surjective in degree2:

H 2c .Hilb2

0.C2//!H

2.Hilb20.C

2//! f0g:

PROOF. As a matter of fact, if 2Z2�D1

.Hilb20.C

2// is aL2 closed2-form, itsrestriction to the neighborhoodU of infinity4 is exact, according to Proposition3.6. Moreover, forw as in that proposition, we can find2 C1

w;1.U / a primitive

of ˛jU :on U; ˛ D dˇ:

If N 2 C1w;1.Hilb2

0.C2// is an extension of then becausew is a parabolic

weight, ˛ � dˇ and˛ have the sameL2 cohomology class and � dˇ hascompact support. ˜

The Hodge and Poincare dualities imply that we also have an injective map fromL2 cohomology to absolute cohomology:

f0g !H2.Hilb2

0.C2//!H 2.Hilb2

0.C2//:

But the natural map from cohomology with compact support to absolutecohomology is an isomorphism in degree2; hence we have the following iso-morphism:

3L2 cohomology refers to reducedL2

�D1cohomology. From now on we will avoid the subscript1 when

dealing with spaces related toL2

�D1cohomology.

4The ALE condition says that onU , the Eguchi–Hanson metric and the Euclidean metric (for which Uwill be a truncated cone overS3=f˙ Idg) are quasi-isometric. Hence by 2.2, theL2 cohomology of the twometrics are the same.


THEOREM 5.2. For Hilb20.C

2/D T �P1.C/ endowed with the Eguchi–Hansenmetric, we have

Hk.Hilb2

0.C2//D

�

f0g if k 6D 2,R 'H 2

c .Hilb20.C

2//'H 2.Hilb20.C

2// if k D 2.

REMARK 5.3. In fact, there is a general result about theL2 cohomology ofmanifolds with conical ends:Suppose.X n;g/ is a Riemannian manifold withconical ends(meaning that there is a compact set with smooth boundaryK�X

such that.X nK;g/ is isometric to the truncated coneC1.@K/). Letw be asmooth function onX such that

w.r; �/D r�2.1C logr/�2: on X nK ' C1.@K/:

Then, on .X;g/, theL2 range ofd is almost closed in any degree with respecttow and theL2 cohomology of.X;g/ is given by

Hk�.X /D

8

<

:

H kc .X / if k < n=2;

Im�

H kc .X /!H k.X /

�

if k D n=2;

H k.X / if k > n=2:

There are different proofs of this result. The first one uses the scattering calculusdeveloped by Melrose; see [21, Theorem 4] and [14, Theorem 1A]. The seconduses the almost closed range condition, the computation of theL2 and weightedL2w cohomologies of a truncated cone, and the Mayer–Vietoris sequence 3.7

[7, Theorem 4.11]. For the case of the Eguchi–Hanson metric,this topologicalinterpretation of the space ofL2 cohomology can also be obtained using explicitcomputation of harmonic forms because this metric has an SU.2/ invariance;hence the harmonic equation reduces to an ODE [15, section 5.5].

5.2. The case of 3 points

A vanishing result outside degree 4.According to [8], the QALE metric onHilb3

0.C2/ constructed by D. Joyce coincides with the one of H. Nakajima,

who showed in [23] that Hilb30.C2/ can be endowed with a hyperkahler metric

using the hyperkahler reduction of a Euclidean quaternionic space.5 Accordingto N. Hitchin, theL2 cohomology of a hyperkahler reduction of a Euclideanquaternionic space is trivial except perhaps for the degreeequal to the middle(real) dimension [15]. Hence in our case, we only need to compute theL2

cohomology of Hilb30.C2/ in degree4.

5This is a general fact for all the Hilbert schemes of points inC2, Hilbn

0.C2/, n � 2.

26 GILLES CARRON

The result in degree 4.It is again true that for Hilb30.C2/, the natural map from

cohomology with compact support to absolute cohomology is an isomorphismin degree4 and moreover these spaces have dimension1. We have:

LEMMA 5.4. There exists a compact setK � Hilb30.C

2/ such thatHilb30.C

2/

retracts onK and such that anyL2 closed4-form ˛ on Hilb30.C

2/ nK has aprimitiveˇ 2L2

w withwD 1=.r log.rC1//2. In particular, onHilb30.C

2/nK,theL2 range ofd is almost closed in degree4 with respect tow.

Using this and the same arguments as in the case of 2 points, weobtain:

THEOREM 5.5. For Hilb30.C

2/ endowed with the QALE metric described in(4.2)we have:

Hk.Hilb3

0.C2//D

�

f0g if k 6D 4,R 'H 4

c .Hilb30.C

2//'H 4.Hilb30.C

2// if k D 4.

THE PROOF OFLEMMA 5.4. LetK � Hilb30.C

2/ be a compact set such thatHilb3

0.C2/ nK D U [V , where

� U is a truncated cone over.S7=S3/ nO, and� V D f.y; v/ 2 T �P1.C/� .C2 nB2/ W jyj < "jvjg, wherejyj is the pullback

to T �P1.C/ of the Euclidean distance.

The main point in the proof of Lemma 5.4 is the following result concerning theL2 cohomology ofV :

LEMMA 5.6. H4.V / D f0g andZ4.V / D dC3w;1.V /. That is, anyL2 closed

4-form˛ on V has a primitive' 2L2w.�

3V /, that is, ˛ D d'.

We will only sketch the proof of this lemma.Consider 2Z4

�D1.V /. The setU \V is a truncated cone over the product

S3=f˙ Idg � S3 � ."; 2"/. The third Betti number of this product is not zero;hence by Proposition 3.6, there is a 2L2

w.�3.U \V // such that

˛ D d on U \V:

We cannot extend to V as an element ofC3w;1.V / but only as an element

2 C3w;�.V / where�.y; v/D 1= log2.jvjC 1/. Then

˛� d 2Z4�.V /

and, because this form is zero onV \ U , it can be extended to the wholeT �P1.C/ � .C2 n B2/. This extension will be also denoted by� d . Itis still a closed form; that is,

˛� d 2Z4�

�

T �P

1.C/� .C2 nB2/�

:


Now using a Kunneth-type argument and the computation of theL2� cohomology

of C2 nB2, it can be shown that theL2

� cohomology ofT �P1.C/� .C2 nB2/

vanishes in degree4 and that if we introduce the weightw1.y; v/D1=.1Cjyj/2,there are

u 2 C3w;�.T

�P

1.C/� .C2 nB2// and v 2 C3

w1;�.T �

P1.C/� .C2 nB

2//

such that

˛� d D duC dv:

If we let ' D C uC v, then because onV we have"w � w1, we concludethat' 2L2

w.�3V / and

˛ D d':

With this result, we can finish the proof of Lemma 5.4: we cannot use theMayer–Vietoris exact sequence because of the log factor in the weightw (seemiddle of page 20). However, we will use some of the argumentsleading to theproof of the exactness of the Mayer–Vietoris sequence.

Let ˛ be a closedL2 4-form outsideK. BecauseU is a truncated cone, weknow that there is'

U2 C3

w;1.U / such that D d'

Uon U , and by Lemma 5.6,

there is a'V2 C3

w;1.V / such that onV

˛ D d'V :

Now on the intersectionU \V the difference'U�'

Vis a closedL2

w 3-form.But U \ V is a truncated cone overS3=f˙ Idg � S3 � ."; 2"/ and there is ananalogue of Proposition 3.6 for theL2

w cohomology, the threshold now beingn=2�1D 3 in our case. But the second Betti number ofS3=f˙ Idg�S3�."; 2"/

is zero; hence onU \V , 'U�'

Vhas a primitive� 2 C2

w;w.U \V / which canbe extend to a� 2 C2

w;w.U /. Now we can define

D

�

'UC d� on U ,

'V

on V .

By construction, we haveD d and 2L2w

�

�3.Hilb30.C

2/ nK/�

. ˜

5.3. Conclusion.In the physics literature, Hilbn0.C2/ is associated to the moduli

space of instantons on noncommutativeR4 [24]. One motivation for the study of

L2 cohomology of Hilbn0.C2/ comes from a question of C. Vafa and E. Witten:

in [29], see also the nice survey of T. Hausel [13], the following conjecture isformulated (note that2.n� 1/D 1

2dimR Hilbn

0.C2/):

Hk D

�

f0g if k 6D 2.n� 1/;

Im�

H kc .Hilbn

0.C2//!H k.Hilbn

0.C2//�

if k D 2.n� 1/:(5-1)

28 GILLES CARRON

However, Vafa and Witten have said that “unfortunately, we do not understandthe prediction ofS-duality on noncompact manifolds precisely enough to fullyexploit them.” According to N. Hitchin’s vanishing result [15], the first part ofthis conjecture is true. The result above says that this is true fornD 2; 3.

6. Other results and perspectives

In [7], the L2 cohomology of certain QALE spaces is computed. The proofuses the same general idea given in 5.2, but the argumentation is considerablylonger and we cannot in general use the same vanishing result. Quasi Asymptot-ically Locally Euclidean (QALE) geometry is defined by induction. A QALEmanifold asymptotic toCn=� (where� is a finite subgroup of SU.n/) is amanifold whose geometry at infinity is the union of a piece that looks (up to afinite cover) like a subset of the productY �C

p, whereY is a QALE manifoldasymptotic toC

n�p=A for someA a finite subgroup of SU.n � p/. In [7],we computed theL2 cohomology of QALE spaces where the singular spaceC

n=� has only two singular strata (f0g and a finite union of linear subspaces).In order to prove the Vafa–Witten conjecture (5-1), it is important to be ableto understand theL2 cohomology of more general QALE spaces. The almostclosed range condition has been a interesting tool for doingthis for the case ofHilb3

0.C2/. We hope that it will also be useful in other situations and for the

other Hilbert schemes of points.

References

[1] A. Ancona A.Theorie du potentiel sur des graphes et des varietes.Lecture Notesin Math. 1427, Springer, Berlin, 1990.

[2] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Riemanniangeometry I,Math. Proc. Camb. Phil. Soc.77 (1975), 43–69.

[3] E. Calabi, Metriques kahleriennes et fibres holomorphes,Ann. Sci.Ecole Norm.Sup.12:2 (1979), 269–294.

[4] G. Carron, Un theoreme de l’indice relatif.Pacific J. Math.198(2001), 81–107.

[5] G. Carron, Theoremes de l’indice sur les varietes non-compactes,J. Reine Angew.Math., 541(2001), 81–115.

[6] G. Carron,L2-cohomology of manifolds with flat ends,Geom. Funct. Anal.13:2(2003), 366–395.

[7] G. Carron, CohomologieL2 des varietes QALE. arXiv:math.DG/0501290. Toappear inJ. Reine Angew. Math.

[8] G. Carron, On the QALE geometry of Nakajima’s metric, arXiv:0811.3870. Toappear inJ. Inst. Math. Jussieu.


[9] J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds. InGeometry ofthe Laplace operator(Honolulu, 1979). Proc. Sympos. Pure Math., XXXVI, Amer.Math. Soc., Providence, 1980, 91–146,

[10] G. de Rham,Varietes differentiables. Formes, courants, formes harmoniques.3rded., Hermann, Paris, 1973.

[11] T. Eguchi, A. Hanson, Self-dual solutions to Euclideangravity. Ann. Physics,120:1 (1979), 82–106.

[12] A. Grigoryan, Analytic and geometric background of recurrence and non-explo-sion of the Brownian motion on Riemannian manifolds.Bull. Amer. Math. Soc..N.S./36:2 (1999), 135–249.

[13] T. Hausel, S-duality in hyperkahler Hodge theory, preprint, arXiv:0709.0504. Toappear inThe many facets of geometry: a tribute to Nigel Hitchin, edited by O.Garcia-Prada, J.-P. Bourguignon, and S. Salamon, Oxford University Press.

[14] T. Hausel, E. Hunsicker, R. Mazzeo, Hodge cohomology ofgravitational instan-tons,Duke Math. J.122:3 (2004), 485–548.

[15] N. Hitchin, L2-cohomology of hyperkahler quotients,Comm. Math. Phys.211(2000), 153–165.

[16] D. Joyce,Compact manifolds with special holonomy, Oxford University Press(2000) Oxford Mathematical Monographs.

[17] J. Lott. On the spectrum of a finite-volume negatively-curved manifold.Amer. J.Math.123:2 (2001), 185–205.

[18] E. Looijenga,L2-cohomology of locally symmetric varieties,Compositio Math.67 (1998), 3–20.

[19] R. Mazzeo, The Hodge cohomology of a conformally compact metric,J. Differ-ential Geom.28:2 (1988), 309–339.

[20] R. Mazzeo, R. S. Phillips, Hodge theory on hyperbolic manifolds, Duke Math. J.60:2 (1990), 509–559.

[21] R. Melrose, Spectral and scattering theory for the Laplacian on asymptoticallyEuclidean spaces. InSpectral and scattering theory: Proceedings of the Taniguchiinternational workshop, edited by M. Ikawa. Basel: Marcel Dekker. Lect. Notes PureAppl. Math.161(1994), 85–130.

[22] A. Nair, Weighted cohomology of arithmetic groups.Ann. of Math.150:1 (1999),1–31.

[23] H. Nakajima,Lectures on Hilbert schemes of points on surfaces.AmericanMathematical Society, Providence, RI, 1999.

[24] N. Nekrasov, A. Schwarz, Instantons on noncommutativeR4, and.2; 0/ supercon-formal six-dimensional theory.Comm. Math. Phys.198:3 (1998), 689–703.

[25] L. Saper,L2-cohomology of locally symmetric spaces. I,Pure and AppliedMathematics Quarterly, 1:4 (2005), 889–937.

30 GILLES CARRON

[26] L. Saper,L2-harmonic forms on locally symmetric spaces, Oberwolfach reports,2:3, (2005), 2229–2230.

[27] L. Saper, M. Stern, L2-cohomology of arithmetic varieties,Ann. Math.132:1(1990), 1–69.

[28] A. Sen, Dyon-monopole bound states, selfdual harmonicforms on the multi-monopole moduli space and Sl.2;Z/-invariance of string theory,Phys. Lett.B 329(1994), 217–221.

[29] C. Vafa, E. Witten, A strong coupling test ofS -duality.Nuclear Phys. B431:1–2(1994), 3–77.

[30] N. Yeganefar, Sur laL2-cohomologie des varietes a courbure negative, DukeMath. J.122:1 (2004), 145–180.

[31] N. Yeganefar,L2-cohomology of negatively curved Kahler manifolds of finitevolume. Geom. Funct. Anal.15:5 (2005), 1128–1143.

[32] S. Zucker,L2 cohomology of warped products and arithmetic groups,InventionesMath., 70 (1982), 169–218.

GILLES CARRON

LABORATOIRE DE MATHEMATIQUES JEAN LERAY (UMR 6629)UNIVERSITE DE NANTES

2, RUE DE LA HOUSSINIERE

B.P. 9220844322 NANTES CEDEX 3FRANCE

[email protected]


Rigidity of differential operators andChern numbers of singular varieties

ROBERT WAELDER

ABSTRACT. A differential operatorD commuting with anS1-action is saidto be rigid if the nonconstant Fourier coefficients of kerD and cokerD are thesame. Somewhat surprisingly, the study of rigid differential operators turnsout to be closely related to the problem of defining Chern numbers on singularvarieties. This relationship comes into play when we make use of the rigid-ity properties of the complex elliptic genus–essentially an infinite-dimensionalanalogue of a Dirac operator. This paper is a survey of rigidity theorems relatedto the elliptic genus, and their applications to the construction of “singular”Chern numbers.

1. Rigidity of elliptic differential operators

Let D W � .E/! � .F / be an elliptic operator mapping sections of a vectorbundleE to sections ofF . If D commutes with aT D S1 action, then kerDand cokerD are finite-dimensionalS1-modules. We define the character-valuedindex

IndT .D/D kerD� cokerD 2R.T /

For example, ifDDdCd� W˝even!˝odd is the de Rham operator on a smoothmanifoldX with aT action, then by Hodge theory and homotopy invariance ofde Rham cohomology, IndT .D/ is a trivial virtual T -module of rank equal tothe Euler characteristic ofX . In general, if IndT .D/ is a trivial T -module, wesay thatD is rigid. In the case whereD is the de Rham operator, both kerD andcokerD are independently trivialT -modules. However, more interesting casesexist whereD is rigid, but both kerD and cokerD are nontrivialT -modules.For example, ifX is a spin manifold andD W � .�C/! � .��/ is the Dirac

The author is supported by an NSF postdoctoral fellowship.

31

32 ROBERT WAELDER

operator, thenD is rigid. It is instructive to sketch the proof of this fact, whichis due to Atiyah and Hirzebruch [3]:

For simplicity, assume thatT acts onX with isolated fixed pointsfpg, andthat the action lifts to the spin bundles�˙. At each fixed pointp, TpX decom-poses into a sum of one-dimensional complex representations ofT with weightsm1.p/; : : : ;mn.p/, where2n D dimX . If we view IndT .D/ as a function oft 2 T , then by the Lefschetz fixed point formula,

IndT .D/DX

p

1Qn

jD1.tmj =2� t�mj =2/

A priori, IndT .D/ is a function only on the unit circle inC. However, the aboveformula shows that we can analytically continue IndT .D/ to a meromorphicfunction onS2, with possible poles restricted to lie on the unit circle. But sinceIndT .D/ is a virtualT -module, and therefore has a finite Fourier decompositionof the form IndT .D/ D

P

ak tk , all such poles on the unit circle must cancel.It follows that IndT .D/ is constant. Furthermore, by taking the limit ast!1,one sees that the character-valued index is identically zero. A similar proofshows that on a complex manifold,@C @

�(whose corresponding index is the

arithmetic genus) is rigid with respect to holomorphic torus actions.The situation becomes more difficult if we investigate the rigidity of the

twisted Dirac operatorsD˝E, whereE is an equivariant vector bundle. Forexample, ifdS DD˝ .�C˚��/ is the signature operator on a spin manifold,the Lefschetz fixed point formula for the index ofdS ˝TX gives:

IndT .dS ˝TX /DX

p

nY

jD1

1C t�mj .p/

1� t�mj .p/�X

.tmj .p/C t�mj .p//

Here˙mj .p/ are the weights of theT -action on the complexified tangent bun-dle of X at p. The factors

P

.tmj .p/C t�mj .p// come from the twisting of therigid operatordS by TX . Thus, in this situation, the fixed point formula forIndT .dS ˝ TX / has poles at0 and1, and we can no longer apply the sameargument.

It is therefore astonishing that, based on ideas from physics, Witten predictedthe rigidity of an infinite sequence of twisted Dirac operators of this nature on aspin manifold. Witten’s insight came from generalizing a quantum mechanics-inspired proof of the Atiyah–Hirzebruch theorem to its analogue in the settingof superstring theory. We briefly sketch this point of view, as given in [17]:In supersymmetric quantum mechanics on a spin manifoldX (with one realfermion field), the Hilbert space of states corresponds to the space of square-integrable spinors. Quantization of the superchargeQC yields the Dirac op-erator. In passing to superstring theory, the Hilbert spaceof states should be

RIGIDITY AND SINGULAR CHERN NUMBERS 33

interpreted as spinors on the loop space ofX . It therefore makes sense to thinkof the quantization of the supercharge in this theory as a Dirac operator on theloop space. Now for any manifoldX , the loop space ofX possesses a naturalS1 action given by rotating the loops. The fixed points of this action correspondto the space of constant loops, which we may identify withX itself. Via formalapplication of the Atiyah–Bott–Lefschetz fixed point formula one can reducethe S1 character-valued index of operators constructed out ofQC to integralsoverX . To give an example, let� denote the spin bundle on the loop space. Ifwe quantize a theory with two fermionic fields ˙, the associated Hilbert spacebecomes�˝�. Now in finite dimensions,�˝� corresponds to the de Rhamcomplex.�˝� therefore provides a good candidate for the de Rham complexon the loop space. At the classical level of this theory, one has an involution�on the space of superfields, sending C‘� C and �‘ � which preservesthe action Lagrangian. WhenX is spin, this involution descends to the quantumtheory; the corresponding action of� on�˝�may be interpreted as the Hodgestar operator acting on forms. Consequently, one can construct out ofQC and�a canonical choice of a signature operator on the loop space.By the fixed pointformula, itsS1-charactered valued index reduces to the index of

dS ˝

1O

mD1

�qmTX ˝

1O

mD1

SqmTX D dS ˝�q

overX . Here, for any vector bundleE, we define

�qm.E/D 1C qmEC q2mE ^EC � � �

and

Sqm.E/D 1C qmEC q2mE2C � � � ;

whereqm denote the weights of the inducedS1 action of anS1-bundle overX . If X itself has anS1 action, the character-valued index ofdS ˝�q as afunction of ei� may be interpreted as the signatures associated to a family offield theories parameterized by� . The rigidity ofdS ˝�q then follows from aformal application of deformation invariance of the index of Dirac operators onloop spaces. For details, see [16] or [17].

Note that sincedS ˝�q D dS C2qdS ˝TX C� � �, the rigidity ofdS ˝TX

now follows from the rigidity of thedS ˝ �q. It is interesting to point outthat, althoughdS ˝�q is defined on any oriented manifold, it is only rigid forspin manifolds. Heuristically this makes sense when we viewdS ˝�q as thesignature operator on the loop space ofX . For if X is oriented, the signatureoperatordS is easily seen to be rigid. But the the loop space is oriented preciselywhenX is spin.

34 ROBERT WAELDER

Dirac operators on the loop space provide concrete examplesof elliptic gen-era. These are homomorphisms' W˝SO!R from the oriented cobordism ringto an auxiliary ringR, whose characteristic power series are defined in terms ofcertain elliptic integral expressions.

The rigidity theorems of Witten were initially proved underrestricted hy-potheses by Landweber, Stong, and Ochanine [8; 10], and later proved in com-plete form by Bott, Taubes, and Liu [6; 9]. The simplest and most direct proofwas discovered by Liu, who observed that the modular properties of the ellipticgenera implied their rigidity. We will provide a sketch of Liu’s argument for thecase of the complex elliptic genus, which is defined as the index of@˝Eq;y onan almost complex manifold of dimension2n, whereEq;y is given by

Eq;y D y�n=21O

mD1

��yqm�1T 00X ˝��yqmT 0X ˝

1O

mD1

SqmT 00X ˝SqmT 0X

HereTX˝CDT 0X˚T 00X denotes the decomposition of the complexified tan-gent bundle into holomorphic and antiholomorphic components. By Riemann–Roch, the ordinary index of this operator is given by the integral

Z

X

Y

T 0X

xj#�

xj

2� i� z; �

�

#�

xj

2� i; �� :

Here xj denote the formal Chern roots ofT 0X , y D e2� iz and q D e2� i� .#.v; �/ denotes the Jacobi theta function

#.v; �/D

1Y

mD1

.1� qm/ � q1=82 sin�v1Y

mD1

.1� qme2� iv/

1Y

mD1

.1� qme�2� iv/

We will frequently refer to Ind.@˝Eq;y/ as Ell.X I z; �/. The almost-complexversion of Witten’s rigidity theorem for this operator states that the complexelliptic genus ofX is rigid provided thatc1.X /D 0.

The idea of the proof is as follows: For simplicity, assume that theT -actionon X has isolated fixed pointsfpg, with equivariant weightsmj .p/ on T 0

pX .

Writing t 2 T ast D e2� iu, we have by the Lefschetz fixed point formula,

IndT .@˝Eq;y/DX

p

nY

jD1

#.mj .p/u� z; �/

#.mj .p/u; �/

Write IndT .@˝Eq;y/DF.u; z; �/. It is evident from the fixed point formulathatF.u; z; �/ is a meromorphic function onC �C �H which is holomorphic


in z and� . Let z D 1N

whereN is a common multiple of the weightsmj .p/.Then, using the translation formulas:

#.uC 1; �/D�#.u; �/;

#.uC �; �/D q�1=2e�2� iu#.u; �/;

it is easy to see thatF.uC 1; 1N; �/D F.u; 1

N; �/ and thatF.uCN �; 1

N; �/D

F.u; 1N; �/. Thus, for each fixed� , F.u; 1

N; �/ is a meromorphic function on the

torus defined by the latticeZ˚N Z� . Suppose we could show thatF.u; 1N; �/

was in fact holomorphic inu. Then for each multipleN of the weightsmj .p/

and for each� 2H, F.u; 1N; �/ would have to be constant inu. It would follow

that @@u

F.u; 1N; �/ � 0. Since this equation held for an infinite set of.u; z; �/

containing a limit point, it would hold for all.u; z; �/. HenceF.u; z; �/ wouldbe independent ofu, which is precisely the statement of rigidity for the operatorIndT .@˝Eq;y/.

Thus, we are reduced to provingF.u; z; �/ is holomorphic. Let�

ac

bd

�

2

SL2.Z/ act onC �C �H by the rule

.u; z; �/‘� u

c� C d;

z

c� C d;

a� C b

c� C d

�

:

Using the transformation formula

#.u

c� C d;

a� C b

c� C d/D �.c� C d/

1

2 e�icu2

c�Cd #.u; �/;

where� is an eighth root of unity, one sees thatF�

u

c�Cd;

z

c�Cd;

a�Cb

c�Cd

�

isequal to

K �X

p

e�2� icP

n

j D1 mj .p/uz=.c�Cd/nY

jD1

#.mj .p/u� z; �/

#.mj .p/u; �/

whereK is a nonzero holomorphic function of.u; z; �/. Now the Calabi–Yaucondition implies that the only possibleT -action onKX is given by multipli-cation by a constant along the fibers. Since

PnjD1 mj .p/ is the weight of the

T -action induced onK�

X, it follows that

PnjD1 mj .p/ is the same constant for

all p. We may therefore pull the expressione�2� icPn

j D1 mj .p/uz=.c�Cd/ outsideof the above summation sign, and conclude that

F�

u

c�Cd;

z

c�Cd;

a�Cb

c�Cd

�

DK0F.u; z; �/;

for K0 a nonzero holomorphic function.Now the key observation: First, by the fixed point formula,F.u; z; �/ has

possible poles only foruD r C s� , wherer; s 2Q. Moreover, sinceF.u; z; �/

36 ROBERT WAELDER

is the character-valued index of an elliptic differential operator, the poles ofF.u; z; �/ must cancel foru 2 R, since in that caseF.u; z; �/ admits a Fourierdecomposition

P

bk.z; �/e2� iku (in a rigorous treatment of the subject, one

must of course deal with convergence issues regarding this summation). Notethat this is also the key observation in Bott and Taubes’ proof. Thus, foru apossible pole, writeu D .m=`/ .c� C d/, wherec andd are relatively prime.By relative primality, we can find integersa andb so thatad � bc D 1, i.e.,�

ac

bd

�

2 SL2.Z/. Then

K0 �F�

m

`.c� C d/; z; �

�

D F�

m

`;

z

c�Cd;

a�Cb

c�Cd

�

whereK0 ¤ 0. It follows thatF.u; z; �/ is holomorphic, which completes theproof.

The above rigidity theorem for the complex elliptic genus ona Calabi–Yaumanifold has an interesting analogue for toric varieties, which has applicationsto the study of singular varieties. Let be a complete fan which correspondsto a smooth toric varietyX . This means that is a finite union of conesfCig

inside the real vector spaceN ˝ R, whereN is an integral lattice of rankn.For any two conesC1;C2 in ˙ , we require thatC1 \C2 is a proper subcone,and that the union of the cones in covers all ofN ˝ R. The smoothnessrequirement for means that thek-dimensional cones havek generators, eachlying in N . Recall that the data of gives rise to a natural scheme structure asfollows: For each coneC �˙ , we define the sheaf of regular functions

� .UC /D CŒef �f 2SC

whereSC is the collection of linear functionalsf 2 Hom.N;Z/ that are non-negative alongC . The toric varietyX corresponding to these data is the varietywith affine charts given byUC D Specm� .UC /.

Note that inclusions of conesC1 � C2 give rise to inclusions of open setsUC1� UC2

. In particular, since every coneC contains the point0 2 N as asubcone, every open setUC contains the open set

U0 D SpecmCŒeHom.N;Z/�Š .C�/n:

The action of this complex torus on itself is easily seen to extend to all ofUC .In this way,X inherits a natural action by a complex torusTC , with isolatedfixed points.

There is a nice relationship between theTC-invariant divisors on a smoothtoric variety and combinatorial data of its associated simplicial fan: the TC-invariant divisors onX are in one-one correspondence with piecewise linearfunctionals on˙ . For example, iff is a piecewise linear functional on ,thenf is completely determined by its valuesf .vi/ on the generatorsvi 2 N


of the 1-dimensional rays of . These generators, in turn, defineTC-Cartierdivisors by the following prescription: IfC is a cone containingvi , we defineO.vi/.UC / D � .UC / � e

v�i , wherev�

i is the piecewise linear functional whichis 1 on vi and 0 on the remaining1-dimensional rays of . Otherwise, wesetO.vi/.UC / D � .UC /. In this way, each piecewise linearf gives rise tothe divisorDf D

P

f .vi/O.vi/. In terms of this correspondence, it turns outthere is a simple criterion for determining whether aQ-divisor Df is linearlyequivalent to zero: namely,Df �Q 0 if and only if f 2 Hom.N;Q/.

Now, the canonical divisorKX D Df�1, wheref�1 is the piecewise linear

functional given byf�1.vi/ D �1. Clearly if ˙ is complete,f�1 cannot begiven by a globally defined linear functional in Hom.N;Z/. So compact smoothtoric varieties are never Calabi–Yau, and consequently we can expect no rigidityproperties for their complex elliptic genera. Note, however, thatTX is stablyequivalent to

L`iD1 O.vi/, where the sum is taken over all the1-dimensional

raysvi of˙ . Thus, up to a normalization factor, the elliptic genus ofX is givenby the index of@˝ �, where� equals

O

iD1

1O

nD1

��yqn�1O.vi/�1˝��y�1qnO.vi/˝

1O

mD1

SqmO.vi/�1˝SqmO.vi/

We may view� as a function of theTC-line bundle˝`iD1

O.vi/. In this light,is natural to introduce, for anyTC-line bundle˝`

iD1O.vi/

ai , with ai ¤ 0, thefollowing vector bundle, denoted as�.a1; : : : ; a`/:

O

iD1

1O

nD1

��yai qn�1O.vi/�1˝��y�ai qnO.vi/˝

1O

mD1

SqmO.vi/�1˝SqmO.vi/

We may think of@˝�.a1; : : : ; a`/ as a kind of generalized elliptic genus for thetoric varietyX . The analogue of the Calabi–Yau condition for this generalizedelliptic genus is the triviality of theQ-line bundle˝`

iD1O.vi/

ai . In fact, if thisbundle is trivial, then

IndT @˝ �.a1; : : : ; a`/D 0 2R.T /ŒŒq;y;y�1��

for any compact torusT � TC . To prove this, it suffices to assume thatT D S1

and that theT -action onX has isolated fixed points. We can always find sucha T by first picking a dense1-parameter subgroup� of a maximal compactsubtorus ofTC , and then lettingT be generated by a compact1-parameter sub-group whose initial tangent direction is sufficient close tothat of � . Then therigidity of @˝�.a1; : : : ; a`/ follows from Liu’s modularity technique discussedabove. To see that IndT @˝ �.a1; : : : ; a`/ is identically0, we use the following

38 ROBERT WAELDER

trick observed by Hattori [7]. LetF.u; z; �/D IndT @˝�.a1; : : : ; a`/. The mod-ular properties ofF imply thatF.uC �; z; �/D e2� iczF.u; z; �/. Herec is theweight of theT -action on the trivial bundle `

iD1O.vi/

ai . For a generic choiceof T �TC , this weight will be nonzero. But sinceF.u; z; �/ is constant inu, wemust have thatF.u; z; �/D e2� iczF.u; z; �/. This implies thatF.u; z; �/D 0.

2. Chern numbers of singular varieties

We now turn to the problem of defining Chern numbers on singular varieties,a subject which at first glance appears unrelated to the discussion above. In whatfollows we will find that rigidity theorems provide a powerful tool in solvingthese types of problems. We first discuss some background.

If X is a smooth compact almost-complex manifold of dimension2n, theChern numbers ofX are the numbers of the form

ci1;:::;inD

Z

X

ci1

1� c

i2

2� � � cin

n

whereci denotes thei th Chern class ofT 0X andi1C2i2C� � �CninDdimX=2

(so that the total degree of the integrand is2n). Chern numbers are easily seento be functions on the complex cobordism ring�

U. Moreover, they completely

characterize �

Uin the sense that two almost complex manifolds with the same

Chern numbers must be complex cobordant.Much of algebraic geometry consists of efforts to extend techniques from

the theory of smooth manifolds to singular varieties. Minimal model theorysuggests that one can approach this problem by working on a smooth (or “nearlysmooth”) birational model of a given singular varietyX . For a special combi-nation of Chern numbers, this approach works without any difficulties: namely,the Chern numbers defining the Todd genus. For ifX is a smooth complexmanifold, the Todd genus ofX is given by the alternating sum

�0.X /D

nX

iD0

.�1/i dimHi;0

@.X /:

By Hartog’s theorem, the space of holomorphici-forms is birationally invariant,and is therefore well-defined even whenX is singular, by passing to a resolutionof singularities. On the other hand, ifX is smooth, then by Riemann–Roch,

�0.X /D

Z

X

nY

iD1

xi

1� e�xi

wherexi denote the formal Chern roots of the holomorphic tangent bundle. Thecombination of Chern numbers obtained by performing the above integrationtherefore makes sense for any compact singular variety defined overC.


More generally, we consider the following naive attempt at defining combina-tions of Chern numbers onX : Suppose we are lucky enough to have a smoothminimal modelY for X . Then defineci1;:::;in

.X / D ci1;:::;in.Y /. The main

problem with this approach is that, even when smooth minimalmodels exist,we should not expect them to be unique. In fact, we expect different minimalmodels forX to be related to each other by codimension-2 surgeries called flipsand flops. A priori, it is not at all clear what combinations ofChern numberswill be preserved under such operations.

In [11] Totaro set out to classify the combinations of Chern numbers invariantunder classical flops. Here we say that two varietiesX1 and X2 differ by aclassical flop if they are the two small resolutions of ann-fold Y whose singularlocus is locally the product of a smoothn� 3-fold Z and the3-fold nodexy �

zw D 0. More precisely,X1 andX2 are constructed as follows: blowing upalongZ defines a resolution ofY whose exceptional set is aP1 � P1 bundleoverZ with normal bundleO.�1;�1/. HereO.�1;�1/ denotes the line bundleover aP1�P1-bundle which coincides with the tautological bundle alongeachP1 direction. Blowing down along either of theseP1 fibers therefore producestwo distinct small resolutionsX1 andX2 of Y .

Totaro demonstrated that the combinations of Chern numbersinvariant underclassical flops were precisely the combinations of Chern numbers encoded bythe complex elliptic genus in the Riemann–Roch formula. We sketch the firsthalf of his argument–namely, that the complex elliptic genus is invariant underclassical flops. AsX1 andX2 are identical away from their exceptional sets,their differenceX1 � X2 is complex cobordant to a fibrationE over Z. Infact, if the exceptional sets ofX1 andX2 are theP1-bundlesP.A/ andP.B/

corresponding to the rank2 complex bundlesA andB over Z, then as a dif-ferentiable manifold,E is simply theP3 bundle P.A ˚ B�/ over Z. Nowthe way thatE is actually constructed is by taking a tubular neighborhoodofP.A/ � X1 and gluing it to a tubular neighborhood ofP.B/ � X2 along theircommon boundaries (which are both diffeomorphic toZ � S3). The crucialpoint is that the stably almost complex structure onE induced by this construc-tion makesE into an SU-fibration. That is,E is aP3-bundle whose the stabletangent bundle in the vertical direction has a complex structure satisfyingc1D0.He calls these fibers “twisted projective space”zP2;2. The fiber-integration for-mula implies that Ell.EI z; �/ D

R

Z EllT .zP2;2I z; �;x1; : : : ;x4/ � E l l.ZI z; �/.HereE l l.ZI z; �/ is the cohomology class which appears as the integrand inthe Riemann–Roch formula for the elliptic genus ofZ. More importantly,EllT .zP2;2I z; �;x1; : : : ;x4/ denotes the character-valued elliptic genus ofzP2;2

with the standardT 4 action, with the generatorsu1; : : : ;u4 of the Lie algebraof T 4 evaluated at the Chern rootsx1; : : : ;x4 of A˚B. SincezP2;2 is an SU-

40 ROBERT WAELDER

manifold, by the Witten rigidity theorem, EllT .zP2;2I z; �;x1; : : : ;x4/ D const.Thus, the elliptic genus ofE is simply the product Ell.zP2;2I z; �/ �Ell.ZI z; �/.Moreover, sincezP2;2 is cobordant toY1 � Y2, whereYi are the small resolu-tions of a3-fold node, and since classical flopping is symmetric for3-folds,zP2;2 � Y2 �Y1. HencezP2;2 � 0 in the complex cobordism ring. We thereforehave that Ell.X1I z; �/�Ell.X2I z; �/D Ell.zP2;2I z; �/ �Ell.ZI z; �/D 0.

An obvious consequence of the above discussion is that for varietiesY whosesingular locus is locally the product of a smooth variety with a 3-fold node, itmakes sense to define the elliptic genus ofY to be the elliptic genus of one ofits small resolutions. However, most singular varieties fail to possess even onesmall resolution. It is therefore natural to ask whether onecan continue to definethe elliptic genus for a more general class of singularities. The right approachto answering this question is to expand one’s category to include pairs.X;D/,whereX is a variety andD is a divisor onX with the property thatKX �D

is Q-Cartier. A mapf W .X;D/ ! .Y; �/ in this category corresponds to abirational morphismf WX ! Y satisfyingKX �D D f �.KY ��/. The ideais to first define the elliptic genus for smooth pairs.X;D/ in such a way thatEll.X;DI z; �/ becomes functorial with respect to morphisms of pairs. Giventwo resolutionsfi W Xi ! Y of a singular varietyY , with KXi

�Di D f�KY ,

we could then find resolutionsgi W .M;D/ ! .X;Di/ making the followingdiagram commute:

.M;D/g1

��! .X1;D1/

g2

?

?

y

?

?

yf1

.X2;D2/f2

��! .Y; 0/

Functoriality of the elliptic genus would then imply that

Ell.X1;D1I z; �/D Ell.M;DI z; �/D Ell.X2;D2I z; �/:

It would then make sense to define Ell.Y I z; �/� Ell.X1;D1I z; �/:

One can simplify this approach by making two observations. First, by intro-ducing further blow-ups, one can always assume that the exceptional divisorsDi �Xi have smooth components with simple normal crossings. (Suchresolu-tions are called “log resolutions”.) Second, by a deep result of Wlodarczyk [1],the birational map.X1;D1/ 99K .X2;D2/ may be decomposed into a sequenceof maps

.X1;D1/D .X.0/;D.0// 99K � � � 99K .X .N /;D.N //D .X2;D2/

where each of the arrows are blow-ups or blow-downs along smooth centerswhich have normal crossings with respect to the components of D.j/. It there-fore suffices to define Ell.X;DI z; �/ for smooth pairs.X;D/, whereD is a


simple normal crossing divisor, and prove that Ell.X;DI z; �/ is functorial withrespect to blow-ups along smooth centers which have normal crossings with re-spect to the components ofD. This procedure has been carried out successfullyby Borisov and Libgober in [4], and by Chin-Lung Wang in [15].They defineEll.X;DI z; �/ by the formula

Z

X

Y

j

xj#�

xj

2� i� z; �

�

#�

xj

2� i; ��

Y

i

#�

Di

2� i� .ai C 1/z; �

�

#.z; �/

#�

Di

2� i� z; �

�

#..aiC 1/z; �/(2-1)

Here thexj denote the formal Chern roots ofTX and theDi denote the firstChern classes of the componentsDi of D with coefficientsai.X;D/. Note thatsince#.0; �/D0, the above expression only makes sense forai¤�1. Naturally,this places some restrictions on the types of singularitiesallowed in the definitionof Ell.Y I z; �/. For example, at the very leastY must possess a log resolution.X;D/ ! .Y; 0/ such that none of the discrepancy coefficientsai.X;D/ areequal to�1. In order to ensure that Ell.Y I z; �/ does not depend on our choiceof a log resolution.X;D/, we actually must require that the discrepancy co-efficientsai.X;D/ > �1. To see why, suppose that.X1;D1/ and .X2;D2/

are two log resolutions ofY with discrepancy coefficientsai.Xj ;Dj / ¤ �1.To prove that Ell.X1;D1I z; �/D Ell.X2;D2I z; �/, we must connect these tworesolutions by a sequence of blow-ups and blow-downs, applying functorialityof the elliptic genus of pairs at each stage. But if some of thediscrepancycoefficientsai.X1;D1/ are greater than�1, and others less than�1, then afterblowing upX1, we may acquire discrepancy coefficients equal to�1. In thiscase, the elliptic genus of one of the intermediate pairs in the chain of blow-upsand blow-downs will be undefined, and consequently we will have no meansof comparing the elliptic genera of.X1;D1/ and.X2;D2/. The only obviousway of avoiding this problem is to requireai.Xj ;Dj / > �1. This constraint isquite familiar to minimal model theorists; singular varieties Y possessing thisproperty are said to havelog-terminalsingularities.

Functoriality of the elliptic genus provides a nice explanation for the invari-ance of the elliptic genus under classical flops. For ifX1 andX2 are relatedby a classical flop, then there exists a common resolutionfi W X ! Xi withf �

1KX1Df �

2KX2

. Two varieties related in this way are said to beK-equivalent.One therefore discovers from this approach that the fundamental relation leavingthe elliptic genus invariant is not flopping butK-equivalence.

The original proofs of functoriality of the elliptic genus,by Borisov, Lib-gober, and Wang, are based on an explicit calculation of the push-forwardf�

of the integrand in (2-1), wheref W .X;D/! .X0;D0/ is a blow-down. Theobstruction to this push-forward giving the correct integrand onX0 is given

42 ROBERT WAELDER

by an elliptic function with values inH �.X0/. One can then use basic ellipticfunction theory to show that this function vanishes. In whatfollows, we willsketch a different proof, similar to the one in [13], that makes use of the rigidityproperties of the elliptic genus. This approach has severaladvantages: the firstis that the proof can be easily generalized to more exotic versions of ellipticgenera, such as the character-valued elliptic genus for orbifolds. Though theoriginal proofs could be adapted to this situation, their implementation in themost general setting is cumbersome. Another advantage is that some variationof this approach appears to be useful for studying elliptic genera for varietieswith non-log-terminal singularities. We will have more to say on this in thefollowing section. Recall though that the rigidity of the elliptic genus for SU-manifolds was the key step in Totaro’s proof of the invariance of elliptic generaunder classical flops. It is therefore reasonable to expect rigidity phenomena toplay a useful role in the study of elliptic genera of singularvarieties.

Proceeding with the proof, we letX be a smooth variety andD DP

aiDi asimple normal crossing divisor onX . Let f W zX ! X be the blow-up along asmooth subvariety which has normal crossings with respect to the componentsof D. We let zD D

P

aizDi CmE be the sum of the proper transforms ofDi

and the exceptional divisorE, whose coefficients are chosen so thatK zX� zDD

f �.KX �D/.To avoid getting bogged down in technical details, assumef W zX !X is the

blow-up at a single pointpDD1\ : : :\Dn, andD1; : : : ;Dn are the only com-ponents ofD. ThenT zX is stably equivalent tof �TX˚

LniD1 O.

zDi/˚O.E/.From (2-1), it follows immediately that the proof of the blow-up formula for theelliptic genus amounts to proving that

Z

zX

f �

�

Y

T 0X

xj

2� i#.

xj

2� i� z/# 0.0/

#.xj

2� i/#.�z/

� nY

iD1

zDi

2� i#.

zDi

2� i� .ai C 1/z/# 0.0/

#.zDi

2� i/#.�.ai C 1/z/

�

E2� i

#. E2� i� .mC 1/z/# 0.0/

#. E2� i

/#.�.mC 1/z/

D

Z

X

Y

T 0X

xj

2� i#.

xj

2� i� z/# 0.0/

#.xj

2� i/#.�z/

nY

iD1

Di

2� i#. Di

2� i� .ai C 1/z/# 0.0/

#. Di

2� i/#.�.aiC 1/z/

Here, for ease of exposition, we have omitted the dependenceof # on � . Notethat zDi D f

�Di �E in the above expression. Thus, if we expand both sidesin the variablesf �Di ;E, andDi , the blow-up formula is easily seen to holdfor integrals of Chern and divisor data not involvingE. Note however that in aneighborhood ofE, . zX ; zD/ has the exact same structure as the blow-up ofC

n atthe origin, with the divisorszD1; : : : ; zDn corresponding to the proper transforms


of the coordinate hyperplanes ofCn. For the purpose of proving the blow-up

formula, we may therefore assume thatX Š .P1/n and that zX is the blow-upof X at Œ0 W 1�� Œ0 W 1�. Viewed as a toric variety,X is defined by the fan˙ � N ˝R with 1-dim raysR.˙e1/; : : : ;R.˙en/, wheree1; : : : ; en form anintegral basis for the latticeN . The fan z of zX is obtained from by addingthe rayR.e1C� � �Cen/. The divisorsDi �X correspond to the raysRei in ˙ ;and the divisorszDi andE correspond to the raysRei andR.e1C� � �Cen/ in z .Using the fact that the tangent bundle of smooth toric variety with TC-invariantdivisors Dj , j D 1; : : : ; `, is stably equivalent to

L`jD1 O.Dj /, the blow-up

formula forX reduces to proving that

Z

zX

nC1Y

kD1

zDk

2� i#.

zDk

2� i� .ak C 1/z/# 0.0/

#.zDk

2� i/#.�.ak C 1/z/

nY

kD1

zD�k

2� i#.

zD�k

2� i� .a�k C 1/z/# 0.0/

#.zD�k

2� i/#.�.a�k C 1/z/

D

Z

X

nY

jD1

Dj

2� i#.

Dj

2� i� .aj C 1/z/# 0.0/

#.Dj

2� i/#.�.aj C 1/z/

nY

jD1

D�j

2� i#.

D�j

2� i� .a�j C 1/z/# 0.0/

#.D�j

2� i/#.�.a�j C 1/z/

In this formula,D�j denote theTC-divisors onX corresponding to the1-dimraysR.�ej /, with coefficientsa�j D 0. zD�j denote their proper transforms,which are simply given byf �D�j , sinceD�j are defined away from the blow-up locus. For ease of exposition, we also letzDnC1 DE, with anC1 Dm.

Now the crucial observation is that in the above formula, RHS� LHS isindependent of the coefficientsa�j . For since zD�j are disjoint fromE, anydivisor intersection data involvingzD�j will be unchanged after formally settingE D 0. Therefore, the parts of RHS� LHS dependinga�j will be unchangedafter settingE D 0. But formally lettingE D 0 clearly gives RHSD LHS.Consequently, RHS�LHS depends only ona1; : : : ; an.

Let us therefore definea�j so that.1Ca�j /D�.1Caj /. As discussed in theprevious section, the set of coefficients.1C a˙j / assigned to the raysR.˙ej /

give rise to a piecewise linear functionalg D g1Cai ;1Ca�ion the fan˙ . In

fact,g is simply the global linear functional which maps the basis vectorsei to.1C ai/. As g 2 Hom.N;Z/, it also defines a global linear functional onz ,taking the value

PniD1.1Cai/ one1C� � �Cen. Now by the discrepancy formula

for blow-ups,Pn

iD1.1Cai/D .1Cm/. We see from this that the piecewise linearfunctional on z defined by assigning the coefficients.1Ca˙j / to R.˙ej / and.1Cm/ to R.e1C� � �Cen/ corresponds to this same global linear functionalg.

It follows that the bundles

O.e1C � � �C en/1Cm˝

nO

iD1

O.ei/1Cai ˝O.�ei/

1Ca�i

44 ROBERT WAELDER

andNn

iD1 O.ei/1Cai ˝O.�ei/

1Ca�i are trivial asQ-line bundles onzX andX , respectively. Consequently,

Ind @˝ �.1C ai ; 1Cm; 1C a�i/D Ind @˝ �.1C ai ; 1C a�i/D 0:

But, up to a normalization factor, Ind@˝ �.1C ai ; 1Cm; 1C a�i/ D RHSand Ind@˝ �.1C ai ; 1C a�i/D LHS for the given new values ofa�i . Thus,RHSD LHS for .1C a�i/D�.1C ai/, and therefore also fora�i D 0.

This completes the proof of the blow-up formula for the case where the blow-up locus is a single point. For completeness, let us outline the case for the blow-up along a subvarietyZ with normal crossings with respect to the componentsof D. This case is handled in much the same way, the only difference being thatinstead of reducing to the situation whereX is toric, we instead reduce to thecase whereX is a toric fibration, fibered over the blow-up locusZ. Namely, bydeformation to the normal cone, we may assume that

X D P.M ˚ 1/�P.L1˚ 1/� � � � �P.Lr ˚ 1/:

Here, for the componentsDi intersectingZ,

Li DO.Di/jZ

andM is the quotient ofNZ=X by˚Li . The product� is the fiber product ofthe corresponding projective bundles overZ. We now viewDi as the divisorsgiven by the zero sections of the bundlesLi . Moreover, the zero sections ofLi

andM together define a copy ofZ in P.M ˚1/�P.L1˚1/�� P.Lr ˚1/

with the same normal bundleNZ=X as in the original space. We letzX be theblow-up along this copy ofZ. The proof of the blow-up formula then followsthe same reasoning as in the toric case, where we now make use of the rigidityof fiberwise analogues of the operators@˝ �.Ea/. For example, let us examinehow to generalize the bundle�.1C ai ; 1C a�i/ on .P1/n to the fibrationX .

For each fibration�i W P.Li ˚ 1/!Z, we have the exact sequence

0! Si! ��

i .Li ˚ 1/!Qi ! 0

of tautological bundles. The vertical tangent bundle toP.Li ˚ 1/ is stablyequivalent to the direct sum of hyperplane bundlesHi ˚ H�i , whereHi D

Hom.��

i Li ;Si/ andH�i D Hom.1;Si/. Similarly, the vertical tangent bundleto the fibration

� W P.M ˚ 1/!Z;

with tautological bundleS is stably equivalent to the direct sumV ˚H whereV D Hom.��M;S/ andH D Hom.1;S/. All of these bundles extend natu-rally to the whole fibrationX . Recall that if˛i D �˛�i , then@˝ �.˛i ; ˛�i/

defines a elliptic operator on.P1/n with vanishing equivariant index (note that


for convenience of notation we have defined˛i D 1C ai). For the fibrationX ,we replace@˝ �.˛i ; ˛�i/ by the following fiberwise analogue:

@˝

˙rO

iD˙1

1O

nD1

��y˛i qn�1H �

i ˝�y�˛i qnHi ˝

1O

mD1

SqmH �

i ˝SqmHi

˝

1O

nD1

��yqn�1V �˝�y�1qnV ˝

1O

mD1

SqmV �˝SqmV

˝

1O

nD1

��y�d�1qn�1H �˝�ydC1qnH ˝

1O

mD1

SqmH �˝SqmH:

Hered D rank.M /. By performing a fiber integration overX , one can showthat the rigidity of this operator with respect to the obvious torus action onthe fibers follows directly from the rigidity results obtained for@˝ �.˛i ; ˛�i/.Analogously, there exists a natural generalization of the operator@ ˝ �.1 Cai ; 1Cm; 1C a�i/ to a rigid operator on the fibrationzX . We therefore seethat the blow-up formula for the elliptic genus is in all cases a consequence ofrigidity phenomena on toric varieties.

Before moving on, we make a simple observation which will prove conve-nient in the next section. LetX be a smooth toric variety with toric divisorsDi .SinceTX is stably equivalent to

L`iD1 O.Di/, the elliptic genus of the pair

.X;P

aiDi/ is equal to the index of the operator

@˝ �.a1C 1; : : : ; a`C 1/;

up to a normalization factor. Moreover, the condition thatN`

iD1 O.Di/ai C1 is

trivial is equivalent to the condition thatKX �P

aiDi D 0 as a Cartier divisor.In this case, we say that.X;

P

aiDi/ is a Calabi–Yau pair. Hence, a trivialconsequence of the rigidity theorem for the elliptic genus of toric varieties isthat Ell.X;DI z; �/D 0 whenever.X;D/ is a toric Calabi–Yau pair.

3. Beyond log-terminal singularities

As observed above, Borisov, Libgober, and Wang’s approach to defining theelliptic genus of a singular varietyY only appears to work whenY has log-terminal singularities. This is due to the division by#..aiC1/z/ in the formulafor the elliptic genus of the pair.X;D/, where.X;D/ is a resolution ofY withdiscrepancy coefficientsai.X;D/. In pursuit of the broader question, “for whatclass of singularities can we make sense of Chern data?”, it is natural to askwhether log-terminality represents an essential constraint. In what follows, wewill demonstrate that at the very least, the elliptic genus can be defined for allbut a finite class of normal surface singularities.

46 ROBERT WAELDER

Since the terms#..aiC1/z/ do not involve any Chern data, the first thing onemight try to do is simply throw these terms away in the definition of the ellipticgenus of a pair. However, this approach is of little use sinceone would losefunctoriality with respect to birational morphisms. As a second attempt, onecould introduce a perturbationai C "bi to each of the discrepancy coefficientsai of D, and take the limit as"! 0. Two obvious difficulties with this approachare.1/ the limit does not always exist, and.2/, even when the limit exists, itdepends on the choice of the perturbation. Moreover, deciding on some fixedperturbation in advance (like letting allbi D 1) runs into problems if we hopeto preserve functoriality.

To carry out this perturbation approach, we therefore require a distinguishedclass of perturbation divisors�.X;D/ D f

P

"biDig satisfying the followingtwo properties:

(1) For everyD" 2�.X;D/, the limit as"! 0 of Ell.X;DCD"I z; �/ existsand is independent of the choice ofD".

(2) If f W . zX ; zD/! .X;D/ is a blow-up, thenf ��.X;D/��. zX ; zD/.

Assuming we have found a set of perturbation divisors satisfying these proper-ties, we could then define the elliptic genus of a singular variety Y by the follow-ing procedure: Pick a log-resolution.X;D/ of Y , and chooseD" 2 �.X;D/.Then define Ell.Y I z; �/ D lim"!0 Ell.X;D CD"I z; �/. The important pointis that if f W . zX ; zD/ ! .X;D/ is a blow-up, andzD" 2 �. zX ; zD/, then theanswer we get for the elliptic genus ofY is the same, regardless of whetherwe work with .X;D C D"/ or with . zX ; zD C zD"/. To see why, note thatf �.KX �D �D"/ D K zX

� zD � f �D". Thus, by functoriality of the elliptic

genus with respect to blow-ups, Ell.X;DCD"I z; �/DEll. zX ; zDCf �D"I z; �/.By property.2/, f �D" lies inside�. zX ; zD/. Hence, property.1/ of �. zX ; zD/implies that lim"!0 Ell. zX ; zDC zD"I z; �/D lim"!0 Ell. zX ; zDC f �D"I z; �/.

For the case of complex surfaces, we have a natural candidatefor �.X;D/satisfying the second property; namely the set

f�" W�"Di D 0 for all Di �D with discrepancy coefficientD�1g

For if�" is in this set, andzDi � zD has coefficient equal to�1, thenf ��" zDi D

�"f�zDi . Now, either zDi is the proper transform of a divisor with�1 discrep-

ancy, or it is a component of the exceptional locus off . In the former case,�"f�

zDi D 0 by virtue�" belonging to the set�.X;D/; in the latter case,f�zDi D 0.We still must verify that the"!0 limit of Ell .X;DCD"I z; �/ is well-defined

and independent of the choice ofD" 2 �.X;D/ when .X;D/ is a resolutionof a singular complex surfaceY . Unfortunately, it is too much to ask that this


property hold for all normal surface singularities. Suppose, however, that.X;D/is a log resolution of a normal surfaceY satisfying the following additionalproperty: For every componentDi�D with discrepancy coefficientai.X;D/D

�1, Di Š P1 andDi intersects at most two other componentsDi1;Di2

of D

at a single point (andaik.D/ ¤ �1 for k D 1; 2). In other words, we assume

that the local geometry in a tubular neighborhoodU of Di is indistinguishablefrom a tubular neighborhood of a toric divisor, and moreoverthat we can finddisjoint such neighborhoods for every componentDi with a �1 discrepancy.Note that sinceDi is an exceptional curve, the adjunction formula implies that.X;D/jU is a toric Calabi–Yau pair. Under this additional assumption, it turnsout that lim"!0 Ell.X;DCD"I z; �/ exists and is independent of the choice ofD" 2�.X;D/.

To see why the limit exists, note that Ell.X;D C D"I z; �/ is a meromor-phic function of " with at most a simple pole at" D 0. Up to a normal-ization factor, the residue of Ell.X;D C D"I z; �/ at " D 0 corresponds toP

ai .X ;D/D�1 Ell.Di ;DCD"jDiI z; �/. By adjunction,.Di ;DCD"jDi

/ are alltoric Calabi–Yau pairs, and consequently, the residue of Ell.X;D CD"I z; �/

vanishes by the rigidity theorems discussed in the previoussection.It remains to check that this limit is independent of the choice of D" 2

�.X;D/. Suppose then thatD";D0" are two possible perturbation divisors.

Since the"! 0 limit of Ell .X;DCD"I z; �/�Ell.X;DCD0"I z; �/ depends

only on the local geometry near the divisor componentsDi with ai.X;D/D�1,we may reduce the calculation to the case where.X;D/ is a toric variety. More-over, since.X;D/ is Calabi–Yau in the tubular neighborhoodsUi of the abovedivisor components, we may further reduce to the situation where.X;D/ isa Calabi–Yau pair. By definition,D" andD0

" are trivial overUi and we mayextend them to trivial divisors over all ofX without affecting the"! 0 limitof Ell.X;D C D"I z; �/ or Ell.X;D C D0

"I z; �/ . We have thus reduced thecalculation to the case where.X;D C D"/ and .X;D C D0

"/ are both toricCalabi–Yau pairs. The rigidity theorem for the elliptic genus in this case thenimplies that Ell.X;DCD"I z; �/ D Ell.X;DCD0

"I z; �/ D 0 for all ", whichclearly implies that their limits are the same as"! 0.

Of course, the above discussion is moot unless one can find a reasonablylarge class of surface singularities whose resolutions satisfy the additional crite-rion of being locally toric in a neighborhood of the exceptional curves with�1

discrepancies. Fortunately, as observed by Willem Veys [12], nearly all normalsurface singularities satisfy this property. The sole exceptions consist of the nor-mal surfaces with strictly log-canonical singularities. These are surfaces whoseresolutions.X;D/ satisfyai.X;D/� �1, with someai.X;D/D�1. A well-known example is the surface singularity obtained by collapsing an elliptic curve

48 ROBERT WAELDER

to a point. For a complete classification of these singularities, see [2]. Based onthis observation, Veys used a limiting procedure similar tothe one given hereto define Batyrev’s string-theoretic Hodge numbers for normal surfaces withoutstrictly log-canonical singularities.

Note that, for dimensionality reasons, the elliptic genus of a smooth surfaceis a coarser invariant than the surface’s collective Hodge numbers. Nevertheless,the approach discussed here affords several advantages. First, the technique ofapplying the rigidity properties of toric Calabi–Yau pairsis easy to adapt tomore complicated invariants, such as the character-valuedelliptic genus and theelliptic genus of singular orbifolds. These are finer invariants than the ordinaryelliptic genus which are not characterized entirely by Hodge numbers. Second,this approach provides some clues about how to define elliptic genera for higher-dimensional varieties whose singularities are not log-terminal. For example, apossible generalization of the locally toric structure we required of the�1 dis-crepancy curves is to demand that all�1 discrepancy divisors be toric varietiesfibered over some smooth base. The analogue of property.2/ for �.X;D/ inthis case is thatc1.D"/ D 0 when restricted to each fiber of a�1 discrepancydivisor.

4. Further directions

Singular Chern numbers constructed out of elliptic genera have an interestinginterpretation when the singular variety is the quotient ofa smooth varietyXby a finite groupG. In this situation, quantum field theory on orbifolds givesrise to a definition for the elliptic genus ofX=G constructed entirely out of theorbifold data of.X;G/. This orbifold version of the elliptic genus turns outto be closely related to the singular elliptic genus ofX=G: for example, whenthe G-action has no ramification divisor, the orbifold elliptic genus of.X;G/equals the singular elliptic genus ofX=G. This is a specific example, proved byBorisov and Libgober [5], of a much larger interaction between representationtheory and topology known as the McKay correspondence.

Note that the log-terminality constraint comes for free in this case, sincethe germs of quotientsCn=G, whereG is a finite subgroup ofGL.n;C/ arealways log-terminal. Suppose however thatX itself is singular. By followinga procedure similar to the one discussed above for the elliptic genus, one cancontinue to define a singular analogue of the orbifold elliptic genus of.X;G/.At this point it is natural to ask whether the McKay correspondence continuesto hold when we allowX to have singularities. WhenX has log-terminal singu-larities, this follows directly out of Borisov and Libgober’s proof of the McKaycorrespondence. For more general singularities the answerto this question isnot known, although the McKay correspondence has been verified for the cases


discussed in the previous section: namely, whenX is a normal surface withoutstrictly log-canonical singularities. See, for example, [14].

As we have seen, many of the techniques for studying ellipticgenera in bi-rational geometry can be traced back to some rigidity property of the ellipticgenus. It is therefore not surprising that most of these techniques (such asfunctoriality of the elliptic genus of a divisor pair) work equally well for thecharacter-valued elliptic genus. From Totaro’s work, we know that the ellipticgenus completely determines the collection of Chern numbers invariant underclassical flops. An obvious question then is whether the analogous statementholds for equivariant Chern numbers. From the functoriality property of thecharacter-valued elliptic genus, one easily verifies that the equivariant Chernnumbers encoded by the character-valued elliptic genus areindeed invariantunder equivariant flops. The more difficult question is whether all flop-invariantequivariant Chern numbers factor through the character-valued elliptic genus. Itappears that some knowledge of the image of the character-valued elliptic genusover an equivariant cobordism ring must play a role in answering this question.

References

[1] Dan Abramovich, Kalle Karu, Kenji Matsuki, and JaroslawWlodarczyk. “Torifica-tion and factorization of birational maps.”J. Amer. Math. Soc., 15(3):531–572, 2002.

[2] “Flips and abundance for algebraic threefolds”. InPapers from the Second SummerSeminar on Algebraic Geometry held at the University of Utah, (Salt Lake City,1991), Asterisque No. 211, Societe Mathematique de France, Paris, 1992.

[3] Michael Atiyah and Friedrich Hirzebruch. “Spin-manifolds and group actions.” InEssays on Topology and Related Topics.Memoires dedies a Georges de Rham/.Springer, New York, 1970, 18–28.

[4] Lev Borisov and Anatoly Libgober. “Elliptic genera of singular varieties.”DukeMath. J., 116(2):319–351, 2003.

[5] Lev Borisov and Anatoly Libgober. “McKay correspondence for elliptic genera.”Ann. of Math. (2), 161(3):1521–1569, 2005.

[6] Raoul Bott and Clifford Taubes. “On the rigidity theorems of Witten.” J. Amer.Math. Soc., 2(1):137–186, 1989.

[7] Akio Hattori. “Elliptic genera, torus orbifolds and multi-fans. II.” Internat. J. Math.,17(6):707–735, 2006.

[8] Peter S. Landweber and Robert E. Stong. “Circle actions on spin manifolds andcharacteristic numbers.”Topology, 27(2):145–161, 1988.

[9] Kefeng Liu. “On elliptic genera and theta-functions.”Topology, 35(3):617–640,1996.

[10] Serge Ochanine. “Genres elliptiquesequivariants.” InElliptic curves and modularforms in algebraic topology, 107–122. Springer, Berlin, 1988.

50 ROBERT WAELDER

[11] Burt Totaro. “Chern numbers of singular varieties and elliptic homology.” Ann. ofMath..2/, 151(2):757–791, 2000.

[12] Willem Veys. “Stringy invariants of normal surfaces.”J. Algebraic Geom., 13(1):115–141, 2004.

[13] Robert Waelder. “Equivariant elliptic genera.”Pacific J. Math., 235(2):345–377,2008.

[14] Robert Waelder. “Singular McKay correspondence for normal surfaces.” Preprint,math.AG/0810.3634.

[15] Chin-Lung Wang. “K-equivalence in birational geometry and characterizations ofcomplex elliptic genera.”J. Algebraic Geom., 12(2):285–306, 2003.

[16] Edward Witten. “The index of the Dirac operator in loop space.” InElliptic curvesand modular forms in algebraic topology, 161–181. Springer, Berlin, 1988.

[17] Edward Witten. “Index of Dirac operators.” InQuantum fields and strings: acourse for mathematicians, Vol. 1, 475–511. Amer. Math. Soc., Providence, RI,1999.

ROBERT WAELDER

DEPARTMENT OFMATHEMATICS, STATISTICS, AND COMPUTER SCIENCE

UNIVERSITY OF ILLINOIS AT CHICAGO

410 SCIENCE AND ENGINEERING OFFICES

851 S. MORGAN STREET

CHICAGO, IL [email protected]


Hodge theory meets the minimal modelprogram: a survey of log canonical and

Du Bois singularitiesSANDOR J. KOVACS AND KARL E. SCHWEDE

ABSTRACT. We survey some recent developments in the study of singulari-ties related to the classification theory of algebraic varieties. In particular, thedefinition and basic properties of Du Bois singularities andtheir connectionsto the more commonly known singularities of the minimal model program arereviewed and discussed.

1. Introduction

The primary goal of this note is to survey some recent developments in thestudy of singularities related to the minimal model program. In particular, wereview the definition and basic properties ofDu Bois singularitiesand explainhow these singularities fit into the minimal model program and moduli theory.

Since we can resolve singularities [Hir64], one might ask why we care aboutthem at all. It turns out that in various situations we are forced to work withsingularities even if we are only interested in understanding smooth objects.

One reason we are led to study singular varieties is providedby the minimalmodel program [KM98]. The main goal is the classification of algebraic vari-eties and the plan is to find reasonably simple representatives of all birationalclasses and then classify these representatives. It turns out that the simplestobjects in a birational class tend to be singular. What this really means is thatwhen choosing a birational representative, we aim to have simpleglobal prop-erties and this is often achieved by a singular variety. Being singular means that

Mathematics Subject Classification:14B05.

Kovacs was supported in part by NSF Grant DMS-0554697 and the Craig McKibben and Sarah MernerEndowed Professorship in Mathematics. Schwede was partially supported by RTG grant number 0502170and by a National Science Foundation Postdoctoral ResearchFellowship.

51

52 SANDOR J. KOVACS AND KARL E. SCHWEDE

there are points where thelocal structure is more complicated than on a smoothvariety, but that allows for the possibility of still havinga somewhat simplerglobal structure and along with it, good local properties atmost points.

Another reason to study singularities is that to understandsmooth objects weshould also understand how smooth objects may deform and degenerate. Thisleads to the need to construct and understand moduli spaces.And not just modulifor the smooth objects: degenerations provide important information as well. Inother words, it is always useful to work with complete moduliproblems, i.e.,extend our moduli functor so it admits a compact (and preferably projective)coarse moduli space. This also leads to having to consider singular varieties.

On the other hand, we have to be careful to limit the kinds of singularities thatwe allow in order to be able to handle them. One might view thissurvey as alist of the singularities that we must deal with to achieve the goals stated above.Fortunately, it is also a class of singularities with which we have a reasonablechance to be able to work.

In particular, we will review Du Bois singularities and related notions, in-cluding some very recent important results. We will also review a family ofsingularities defined via characteristic-p methods, the Frobenius morphism, andtheir connections to the other set of singularities we are discussing.

Definitions and notation. Let k be an algebraically closed field. Unless oth-erwise stated, all objects will be assumed to be defined overk. A schemewillrefer to a scheme of finite type overk and unless stated otherwise, apoint refersto a closed point.

For a morphismY ! S and another morphismT ! S , the symbolYT willdenoteY �S T . In particular, fort 2 S we writeXt D f

�1.t/. In addition, ifT D SpecF , thenYT will also be denoted byYF .

Let X be a scheme andF anOX -module. Them-th reflexive powerof F isthe double dual (or reflexive hull) of them-th tensor power ofF :

FŒm� WD .F˝m/��:

A line bundleon X is an invertibleOX -module. AQ-line bundleL on X isa reflexiveOX -module of rank1 that possesses a reflexive power which is aline bundle, i.e., there exists anm 2 NC such thatL Œm� is a line bundle. Thesmallest suchm is called theindexof L .

� For the advanced reader: whenever we mention Weil divisors,assume thatXis S2 [Har77, Theorem 8.22A(2)] and think of aWeil divisorial sheaf, thatis, a rank1 reflexiveOX -module which is locally free in codimension1. Forflatness issues consult [Kol08a, Theorem 2].

� For the novice: whenever we mention Weil divisors, assume thatX is normaland adopt the definition [Har77, p. 130].

A SURVEY OF LOG CANONICAL AND DU BOIS SINGULARITIES 53

For a Weil divisorD onX , its associatedWeil divisorial sheafis theOX -moduleOX .D/ defined on the open setU �X by the formula

� .U;OX .D//D

�

a

b

ˇ

ˇ

ˇ

a; b 2 � .U;OX /; b is not a zero divisor anywhereon U , andDjU C divU .a/�divU .b/� 0:

�

and made into a sheaf by the natural restriction maps.A Weil divisor D on X is aCartier divisor, if its associated Weil divisorial

sheaf,OX .D/ is a line bundle. If the associated Weil divisorial sheaf,OX .D/

is a Q-line bundle, thenD is a Q-Cartier divisor. The latter is equivalent tothe property that there exists anm 2 NC such thatmD is a Cartier divisor.Weil divisors form an abelian group. Tensoring this group with Q (overZ) oneobtains the group ofQ-divisorson X . (If X is not normal, some unexpectedthings can happen in this process; see [Kol92, Chapter 16].)

The symbol� stands forlinear and� for numerical equivalenceof divisors.Let L be a line bundle on a schemeX . It is said to begenerated by global

sectionsif for every pointx 2 X there exists a global section�x 2H 0.X;L /

such that the germ�x generates the stalkLx as anOX -module. IfL is gener-ated by global sections, then the global sections define a morphism

�L WX ! PN D P

�

H 0.X;L /��

:

L is calledsemi-ampleif L m is generated by global sections form� 0. L

is calledample if it is semi-ample and�L m is an embedding form � 0. Aline bundleL on X is calledbig if the global sections ofL m define a rationalmap�L m W X 99K PN such thatX is birational to�L m.X / for m� 0. Notethat in this caseL m need not be generated by global sections, so�L m is notnecessarily defined everywhere. We leave it for the reader the make the obviousadaptation of these notions for the case ofQ-line bundles.

The canonical divisorof a schemeX is denoted byKX and thecanonicalsheafof X is denoted by!X .

A smooth projective varietyX is of general typeif !X is big. It is easy tosee that this condition is invariant under birational equivalence between smoothprojective varieties. An arbitrary projective variety is of general typeif so is adesingularization of it.

A projective variety iscanonically polarizedif !X is ample. Notice that if asmooth projective variety is canonically polarized, then it is of general type.

2. Pairs and resolutions

For the reader’s convenience, we recall a few definitions regarding pairs.


DEFINITION 2.1. A pair .X; �/ consists of a normal1 quasiprojective varietyor complex spaceX and an effectiveQ-divisor� � X . A morphism of pairs W . zX ; z�/! .X; �/ is a morphism W zX ! X such that .Supp.wt�// �Supp.�/. A morphism of pairs W . zX ; z�/ ! .X; �/ is calledbirational ifit induces a birational morphism W wt X z! X and .wt�/ D �. It is anisomorphismif it is birational and it induces an isomorphism W wt X z! X .

DEFINITION 2.2. Let.X; �/ be a pair, andx 2X a point. We say that.X; �/is snc atx, if there exists a Zariski-open neighborhoodU of x such thatU issmooth and�\U is reduced and has only simple normal crossings (see Section3B for additional discussion). The pair.X; �/ is sncif it is snc at allx 2X .

Given a pair.X; �/, let .X; �/reg be the maximal open set ofX where.X; �/ is snc, and let.X; �/Sing be its complement, with the induced reducedsubscheme structure.

REMARK 2.2.1. If a pair.X; �/ is snc at a pointx, this implies that all com-ponents of� are smooth atx. If instead of the condition thatU is Zariski-openone would only require this analytically locally, then Definition 2.2 would definenormal crossing pairs rather than pairs with simple normal crossing.

DEFINITION 2.3. A log resolutionof .X; �/ is a proper birational morphismof pairs� W . zX ;wt�/! .X; �/ that satisfies the following four conditions:

� zX is smooth.� wt�D ��1

�� is the strict transform of�.

� Exc.�/ is of pure codimension1.� Supp. z�[Exc.�// is a simple normal crossings divisor.

If, in addition,

� the strict transformz� of � has smooth support,

then we call� anembedded resolutionof ��X .In many cases, it is also useful to require that� is an isomorphism over

.X; �/reg.

3. Introduction to the singularities of the mmp

Even though we have introduced pairs and most of these singularities makesense for pairs, to make the introduction easier to digest wewill mostly discussthe case when� D ?. As mentioned in the introduction, one of our goals isto show why we are forced to work with singular varieties evenif our primaryinterest lies with smooth varieties.

1Occasionally, we will discuss pairs in the nonnormal setting. See Section 3F for more details.


3A. Canonical singularities. For an excellent introduction to this topic thereader is urged to take a thorough look at Miles Reid’s Young Person’s Guide[Rei87]. Here we will only touch on the subject.

Let us suppose that we would like to get a handle on some varieties. Perhapswe want to classify them or make some computations. In any case, a usefulthing to do is to embed the object in question into a projective space (if we can).Doing so requires a (very) ample line bundle. It turns out that in practice thesecan be difficult to find. In fact, it is not easy to find any nontrivial line bundleon an abstract variety.

One possibility, whenX is smooth, is to try a line bundle that is “handed”to us, namely some (positive or negative) power of thecanonical line bundle;!X D detT �

X. If X is not smooth but instead normal, we can construct!X on

the smooth locus and then push it forward to obtain a rank one reflexive sheafon all of X (which sometimes is still a line bundle). Next we will explore howwe might “force” this line bundle to be ample in some (actually many) cases.

Let X be a minimal surface of general type that contains a.�2/-curve (asmooth rational curve with self-intersection�2). For an example of such asurface consider the following.

EXAMPLE 3.1. zX D .x5C y5C z5Cw5 D 0/ � P3 with the Z2-action thatinterchangesx$y andz$w. This action has five fixed points,Œ1 W1 W�"i W�"i �

for i D 1; : : : ; 5, where" is a primitive fifth root of unity. Hence the quotientzX=Z2 has five singular points, each a simple double point of typeA1. Let

X ! zX=Z2 be the minimal resolution of singularities. ThenX contains five.�2/-curves, the exceptional divisors over the singularities.

Let us return to the general case, that is,X is a minimal surface of general typethat contains a.�2/-curve,C �X . AsC 'P1, andX is smooth, the adjunctionformula gives us thatKX �C D 0. ThereforeKX is not ample.

On the other hand, sinceX is a minimal surface of general type, it followsthat KX is semi-ample, that is, some multiple of it is base-point free. In otherwords, there exists a morphism,

jmKX j WX ! Xcan� P�

H 0.X;OX .mKX //��

:

This may be shown in several ways. For example, it follows from Bombieri’sclassification of pluricanonical maps, but perhaps the simplest proof is providedby Miles Reid [Rei97, E.3].

It is then relatively easy to see that this morphism onto its image is indepen-dent ofm (as long asmKX is base point free). This constant image is calledthecanonical modelof X , and will be denoted byXcan.

The good news is that the canonical line bundle ofXcan is indeed ample, butthe trouble is thatXcan is singular. We might consider this as the first sign of


the necessity of working with singular varieties. Fortunately the singularitiesare not too bad, so we still have a good chance to work with thismodel. In fact,the singularities that can occur on the canonical model of a surface of generaltype belong to a much studied class. This class goes by several names; they arecalleddu Val singularities, or rational double points, or Gorenstein, canonicalsingularities. For more on these singularities, refer to [Dur79; Rei87].

3B. Normal crossings. These singularities already appear in the constructionof the moduli space of stable curves (or if the reader prefers, the constructionof a compactificaton of the moduli space of smooth projectivecurves). If wewant to understand degenerations of smooth families, we have to allow normalcrossings.

A normal crossingsingularity is one that is locally analytically (or formally)isomorphic to the intersection of coordinate hyperplanes in a linear space. Inother words, it is a singularity locally analytically defined as.x1x2 � � �xr D0/�

An for somer � n. In particular, as opposed to the curve case, for surfaces itallows for triple intersections. However, triple intersections may be “resolved”:Let X D .xyzD0/�A3. Blow up the originO 2A3 to obtain� WBlOA3!A3,and consider the proper transform ofX , � W zX ! X . Observe thatzX has onlydouble normal crossings.

Another important point to remember about normal crossingsis that they arenot normal. In particular they do not belong to the previous category. For someinteresting and perhaps surprising examples of surfaces with normal crossingssee [Kol07].

3C. Pinch points. Another nonnormal singularity that can occur as the limit ofsmooth varieties is the pinch point. It is locally analytically defined as the locus.x2

1D x2x2

3/�An. This singularity is a double normal crossing away from the

pinch point. Its normalization is smooth, but blowing up thepinch point (i.e.,the origin) does not make it any better. (Try it for yourself!)

3D. Cones. Let C � P2 be a curve of degreed andX � P3 the projectivizedcone overC . As X is a degreed hypersurface, it admits a smoothing.

EXAMPLE 3.2. Let� D .xd C yd C zd C twd D 0/ � P3xWyWzWw �A1

t . Thespecial fiber�0 is a cone over a smooth plane curve of degreed and the generalfiber�t , for t ¤ 0, is a smooth surface of degreed in P3.

This, again, suggests that we must allow some singularities. The question iswhether we can limit the type of singularities we must deal with. More partic-ularly to this case, can we limit the type of cones we need to allow?

First we need an auxiliary computation. By the nature of the computation itis easier to usedivisorsinstead ofline bundles.


COMMENTARY 3.3. One of our ultimate goals is to construct a moduli spacefor canonical models of varieties. We are already aware thatthe minimal modelprogram has to deal with singularities and so we must allow some singularitieson canonical models. We would also like to understand what constraints areimposed if our goal is to construct a moduli space. The point is that in orderto construct our moduli space, the objects must have an amplecanonical class.It is possible that a family of canonical models degeneratesto a singular fiberthat has singularities worse than the original canonical models. An importantquestion then is whether we may resolve the singularities ofthis special fiberand retain ampleness of the canonical class. The next example shows that thisis not always possible.

EXAMPLE 3.4. LetW be a smooth variety andX D X1 [X2 �W such thatX1 andX2 are Cartier divisors inW . Then by the adjunction formula we have

KX D .KW CX /jX ;

KX1D .KW CX1/jX1

;

KX2D .KW CX2/jX2

:

ThereforeKX jXi

DKXiCX3�i jXi

(3.4.1)

for i D 1; 2, so we have

KX is ample () KX jXiDKXi

CX3�i jXiis ample fori D 1; 2: (3.4.2)

Next, letX be a normal projective surface withKX ample and an isolated sin-gular pointP 2 SingX . Assume thatX is isomorphic to a cone�0 � P3 as inExample 3.2, locally analytically nearP . Further assume thatX is the specialfiber of a family� that itself is smooth. In particular, we may assume that allfibers other thanX are smooth. As explained in (3.3), we would like to seewhether we may resolve the singular pointP 2X and still be able to constructour desired moduli space, i.e., thatK of the resolved fiber would remain ample.For this purpose we may assume thatP is the only singular point ofX .

Let�!� be the blowing up ofP 2� and let zX denote the proper transformof X . Then�0D zX[E, whereE'P2 is the exceptional divisor of the blowup.Clearly,� W zX!X is the blowup ofP onX , so it is a smooth surface andzX\E

is isomorphic to the degreed curve over whichX is locally analytically a cone.We would like to determine the condition ond that ensures that the canonical

divisor of �0 is still ample. According to (3.4.2) this means that we need thatKEC zX jE andK zX

CEj zXbe ample.

As E ' P2, !E ' OP2.�3/, so OE.KE C zX jE/ ' OP2.d � 3/. This isample if and only ifd > 3.


As this computation is local nearP the only relevant issue about the ample-ness ofK zX

CEj zXis whether it is ample in a neighborhood ofEX WDEj zX

. Bythe next claim this is equivalent to asking when.K zX

CEX / �EX is positive.

CLAIM . LetZ be a smooth projective surface with nonnegative Kodaira dimen-sion and� �Z an effective divisor. If .KZC� / �C > 0 for every proper curveC �Z, thenKZ C� is ample.

PROOF. By the assumption on the Kodaira dimension there exists anm > 0

such thatmKZ is effective, hence so ism.KZ C� /. Then by the assumptionon the intersection number,.KZ C � /

2 > 0, so the statement follows by theNakai–Moishezon criterium. ˜

Observe that, by the adjunction formula,

.K zXCEX / �EX D degKEX

D d.d � 3/;

asEX is isomorphic to a plane curve of degreed . Again, we obtain the samecondition as above and thus conclude thatK�0

may be ample only ifd > 3.Now, if we are interested in constructing moduli spaces, oneof the require-

ments of being stable is that the canonical bundle be ample. This means thatin order to obtain a compact moduli space we have to allow conesingularitiesover curves of degreed � 3. The singularity we obtain ford D 2 is a rationaldouble point, but the singularity ford D 3 is not even rational. This does notfit any of the earlier classes we discussed. It belongs to the one discussed in thenext section.

3E. Log canonical singularities.Let us investigate the previous situation undermore general assumptions.

COMPUTATION 3.5. Let D DPr

iD0 �iDi (�i 2 N), be a divisor with onlynormal crossing singularities in a smooth ambient variety such that�0 D 1.Using a generalized version of the adjunction formula showsthat in this situation(3.4.1) remains true.

KD jD0DKD0

C

rX

iD1

�iDi jD0(3.5.1)

Let f W � ! B a projective family with dimB D 1, � smooth andK�b

ample for allb 2 B. Further letX D�b0for someb0 2 B a singular fiber and

let � W � ! � be an embedded resolution ofX � � . Finally let Y D ��X DzX C

PriD1 �iFi where zX is the proper transform ofX andFi are exceptional

divisors for� . We are interested in finding conditions that are necessary for KY

to remain ample.


Let Ei WD Fi j zXbe the exceptional divisors for� W zX ! X and for the

simplicity of computation, assume that theEi are irreducible. ForKY to beample we needKY jwt X as well asKY jFi

for all i to be ample. Clearly, theimportant one of these for our purposes isKY jwt X , for which 3.5.1) gives

KY j zXDK zX

C

rX

iD1

�iEi :

As usual, we may writeK zXD ��KX C

PriD1 aiEi , so we are looking for

conditions to guarantee that��KX CP

.aiC�i/Ei be ample. In particular, itsrestriction to any of theEi has to be ample. To further simplify our computationlet us assume that dimX D 2. Then the condition that we want satisfied is that,for all j ,

� rX

iD1

.ai C�i/Ei

�

�Ej > 0: (3.5.2)

Write the sum in parentheses asEC�E�, where

EC DX

ai C�i �0

jai C�i jEi and E� DX

ai C�i<0

jai C�i jEi :

Choose aj such thatEj � SuppEC. ThenE� �Ej � 0 sinceEj 6�E� and(3.5.2) implies that.EC�E�/ �Ej > 0. These together imply thatEC �Ej > 0

and then thatE2C> 0. However, theEi are exceptional divisors of a birational

morphism, so their intersection matrix,.Ei �Ej / is negative definite.The only way this can happen is ifEC D 0. In other words,ai C �i < 0 for

all i . However, the�i are positive integers, so this implies thatKY may remainample only ifai < �1 for all i D 1; : : : ; r .

The definition of alog canonical singularityis the exact opposite of thiscondition. It requires thatX be normal and admit a resolution of singularities,sayY !X , such that all theai��1. This means that the above argument showsthat we may stand a fighting chance if we resolve singularities that areworsethan log canonical, but have no hope to do so with log canonical singularities.In other words, this is another class of singularities that we have to allow. Aswe remarked above, the class of singularities we obtained for the cones in theprevious subsection belong to this class. In fact, all the normal singularities thatwe have considered so far belong to this class.

The good news is that by now we have covered most of the ways that some-thing can go wrong and found the class of singularities we must allow. Sincewe already know that we have to deal with some nonnormal singularities and infact in this example we have not really needed thatX be normal, we concludethat we will have to allow the nonnormal cousins of log canonical singularities.These are calledsemi-log canonical singularities, and we turn to them now.


3F. Semi-log canonical singularities.Semi-log canonical singularities are veryimportant in moduli theory. These are exactly the singularities that appear onstable varieties, the higher dimensional analogs of stablecurves. However, theirdefinition is rather technical, so the reader might want to skip this section at thefirst reading.

As a warm-up, let us first define the normal and more traditional singularitiesthat are relevant in the minimal model program.

DEFINITION 3.6. A pair.X; �/ is calledlog Q-Gorensteinif KX C� is Q-Cartier, i.e., some integer multiple ofKX C� is a Cartier divisor. Let.X; �/be a logQ-Gorenstein pair andf W zX!X a log resolution of singularities withexceptional divisorE D

S

Ei . Express the log canonical divisor ofzX in termsof KX C� and the exceptional divisors:

K zXCwt�� f �.KX C�/C

X

aiEi

where wt�D f �1� �, the strict transform of� on wtX andai 2 Q. Then the

pair .X; �/ has

terminalcanonical

pltklt

log canonical

9

>

>

>

>

=

>

>

>

>

;

singularities if

8

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

:

ai > 0

ai � 0

ai > �1

ai > �1 andb�c � 0

ai � �1

for all log resolutionsf and alli . The corresponding definitions for nonnormalvarieties are somewhat more cumbersome. We include them here for complete-ness, but the reader should feel free to skip them and assume that for instance“semi-log canonical” means something that can be reasonably considered a non-normal version of log canonical.

Suppose thatX is a reduced equidimensional scheme that

(i) satisfies Serre’s condition S2 (see [Har77, Theorem 8.22A(2)]), and(ii) has only simple normal double crossings in codimension1 (in particularX

is Gorenstein in codimension 1).2

The conditions imply that we can treat the canonical module of X as a divisorialsheaf even thoughX is not normal. Further suppose thatD is aQ-Weil divisoron X (again, following [Kol92, Chapter 16], we assume thatX is regular at thegeneric point of each component in SuppD).

REMARK 3.7. Conditions (i) and (ii) in Definition 3.6 imply thatX is semi-normal since it is seminormal in codimension 1; see [GT80, Corollary 2.7].

2Sometimes a ring that is S2 and Gorenstein in codimension 1 iscalled quasinormal.


Set� WX N !X to be the normalization ofX and suppose thatB is the divisorof the conductor ideal onX N . We denote by��1.D/ the pullback ofD to X N .

DEFINITION 3.8. We say that.X;D/ is semi-log canonicalif

(i) KX CD is Q-Cartier, and(ii) the pair.X N ;BC ��1D/ is log canonical.

Actually, the original definition of semi-log canonical singularities (which isequivalent to this one) uses the theory of semi-resolutions. For details, see[KSB88], [Kol92, Chapter 12], and [Kol08b].

4. Hyperresolutions and Du Bois’s original definition

A very important construction is Du Bois’s generalized De Rham complex.The original construction of Du Bois’s complex, ˝

X, is based on simplicial

resolutions. The reader interested in the details is referred to the original article[DB81]. Note also that a simplified construction was later obtained in [Car85]and [GNPP88] via the general theory of polyhedral and cubic resolutions. At theend of the paper, we include an appendix in which we explain how to construct,and give examples of cubical hyperresolutions. An easily accessible introduc-tion can be found in [Ste85]. Another useful reference is therecent book [PS08].

In [Sch07] one of us found a simpler alternative construction of (part of) theDu Bois complex, not involving a simplicial resolution; seealso Section 6 below.However we will discuss the original construction because it is important to keepin mind the way these singularities appeared, as that explains their usefulness.For more on applications of Du Bois’s complex and Du Bois singularities see[Ste83], [Kol95, Chapter 12], [Kov99], and [Kov00b].

The word “hyperresolution” will refer to either simplicial, polyhedral, or cu-bic resolution. Formally, the construction of˝

Xis the same regardless the type

of resolution used and no specific aspects of either types will be used.The following definition is included to make sense of the statements of some

of the forthcoming theorems. It can be safely ignored if the reader is not inter-ested in the detailed properties of Du Bois’s complex and is willing to acceptthat it is a very close analog of the De Rham complex of smooth varieties.

DEFINITION 4.1. LetX be a complex scheme (i.e., a scheme of finite type overC) of dimension n. LetDfilt .X / denote the derived category of filtered com-plexes ofOX -modules with differentials of order�1 andDfilt ;coh.X / the subcat-egory ofDfilt .X / of complexesK˝ such that for alli , the cohomology sheaves ofGrifilt K˝ are coherent; see [DB81], [GNPP88]. LetD.X / andDcoh.X / denotethe derived categories with the same definition except that the complexes areassumed to have the trivial filtration. The superscriptsC;�; b carry the usual


meaning (bounded below, bounded above, bounded). Isomorphism in thesecategories is denoted by'qis . A sheafF is also considered a complexF ˝

with F 0 DF andF i D 0 for i ¤ 0. If K˝ is a complex in any of the abovecategories, thenhi.K˝/ denotes thei-th cohomology sheaf ofK˝.

The right derived functor of an additive functorF , if it exists, is denoted byRF andRiF is short forhi ıRF . Furthermore,Hi , Hi

Z, andH i

Zwill denote

Ri� , Ri�Z , andRiHZ respectively, where� is the functor of global sections,�Z is the functor of global sections with support in the closed subsetZ, andHZ is the functor of the sheaf of local sections with support in the closed subsetZ. Note that according to this terminology, if� W Y ! X is a morphism andF is a coherent sheaf onY , thenR��F is the complex whose cohomologysheaves give rise to the usual higher direct images ofF .

THEOREM 4.2 [DB81, 6.3, 6.5].Let X be a proper complex scheme of finitetype andD a closed subscheme whose complement is dense inX . Then thereexists a unique object ˝

X2ObDfilt .X / such that, using the notation

˝pXWDGrpfilt ˝

˝

X Œp�;

the following properties are satisfied:

(a) ˝ ˝

X'qis CX ; i.e.,˝ ˝

Xis a resolution of the constant sheafC on X .

(b) ˝ ˝

. / is functorial; i.e., if � W Y ! X is a morphism of proper complexschemes of finite type, there exists a natural map�� of filtered complexes

�� W˝ ˝

X !R��˝˝

Y :

Furthermore, ˝ ˝

X2 Ob

�

Dbfilt ;coh.X /

�

, and if � is proper, �� is a morphismin Db

filt ;coh.X /.

(c) Let U �X be an open subscheme ofX . Then

˝ ˝

X jU 'qis˝˝

U :

(d) If X is proper, there exists a spectral sequence degenerating atE1 andabutting to the singular cohomology ofX :

Epq1DH

q�

X; ˝pX

�

) H pCq.X an;C/:

(e) If "˝WX

˝!X is a hyperresolution, then

˝ ˝

X 'qis R"˝�˝

˝

X˝

:

In particular, hi�

˝pX

�

D 0 for i < 0.

(f) There exists a natural map, OX !˝0X

, compatible with.4:2:.b//.


(g) If X is smooth, then˝ ˝

X 'qis˝˝

X :

In particular,˝

pX'qis˝

pX:

(h) If � W Y ! X is a resolution of singularities, then

˝dimXX 'qis R��!Y :

(i) Suppose that� W zY ! Y is a projective morphism andX � Y a reducedclosed subscheme such that� is an isomorphism outside ofX . Let E denotethe reduced subscheme ofzY with support equal to��1.X / and� 0 WE! X

the induced map. Then for eachp one has an exact triangle of objects in thederived category,

˝pY

// ˝pX˚R��˝

p

zY

�// R� 0

�˝

pE

C1// :

It turns out that Du Bois’s complex behaves very much like thede Rham complexfor smooth varieties. Observe that condition (d) says that the Hodge-to-de Rhamspectral sequence works for singular varieties if one uses the Du Bois complexin place of the de Rham complex. This has far reaching consequences and ifthe associated graded pieces,˝

pX

turn out to be computable, then this singleproperty leads to many applications.

Notice that condition (f) gives a natural mapOX ! ˝0X

, and we will beinterested in situations when this map is a quasi-isomorphism. WhenX is properoverC, such a quasi-isomorphism will imply that the natural map

H i.X an;C/!H i.X;OX /DHi.X; ˝0

X /

is surjective because of the degeneration atE1 of the spectral sequence in con-dition (d).

Following Du Bois, Steenbrink was the first to study this condition and hechristened this property after Du Bois.

DEFINITION 4.3. A schemeX is said to haveDu Bois singularities(or DB sin-gularitiesfor short) if the natural mapOX !˝0

Xfrom condition (f) in Theorem

4.2 is a quasi-isomorphism.

REMARK 4.4. If " WX˝!X is a hyperresolution ofX (see the Appendix for a

how to construct cubical hyperresolutions) thenX has Du Bois singularities ifand only if the natural mapOX !R"

˝�OX˝

is a quasi-isomorphism.

EXAMPLE 4.5. It is easy to see that smooth points are Du Bois. Deligne provedthat normal crossing singularities are Du Bois as well [DJ74, Lemme 2(b)].

We will see more examples of Du Bois singularities in later sections.


5. An injectivity theorem and splitting the Du Bois complex

In this section, we state an injectivity theorem involving the dualizing sheafthat plays a role for Du Bois singularities similar to the role that Grauert–Riemenschneider plays for rational singularities. As an application, we state acriterion for Du Bois singularities related to a “splitting” of the Du Bois complex.

THEOREM 5.1 [Kov99, Lemma 2.2; Sch09, Proposition 5.11].Let X be areduced scheme of finite type overC, x 2 X a (possibly nonclosed) point, andZ D fxg its closure. Assume thatX nZ has Du Bois singularities in a neigh-borhood ofx (for example, x may correspond to an irreducible component ofthe non-Du Bois locus ofX ). Then the natural map

H i�

RHom˝

X .˝0X ; !

˝

X /�

x!H i.!˝

X /x

is injective for everyi .

The proof uses the fact that for a projectiveX , H i.X an;C/! Hi.X; ˝0X/ is

surjective for everyi > 0, which follows from Theorem 4.2.It would also be interesting and useful if the following generalization of this

injectivity were true.

QUESTION 5.2. Suppose thatX is a reduced scheme essentially of finite typeoverC. Is it true that the natural map of sheaves

H i�

RHom˝

X .˝0X ; !

˝

X /�

!H i.!˝

X /

is injective for everyi?

Even though Theorem 5.1 does not answer Question 5.2, it has the followingextremely useful corollary.

THEOREM 5.3 [Kov99, Theorem 2.3; Kol95, Theorem 12.8].Suppose that thenatural mapOX ! ˝0

Xhas a left inverse in the derived category(that is, a

map� W ˝0X! OX such that the compositionOX

// ˝0X

�// OX is an

isomorphism). ThenX has Du Bois singularities.

PROOF. Apply the functorRHomX . ; !˝

X/ to the mapsOX

// ˝0X

�//OX .

Then by the assumption, the composition

!˝

Xı

// RHomX .˝0X; !˝

X/ // !˝

X

is an isomorphism. Letx 2 X be a possibly nonclosed point correspondingto an irreducible component of the non-Du Bois locus ofX and consider thestalks atx of the cohomology sheaves of the complexes above. We obtain thatthe natural map

H i�

RHomX .˝0X ; !

˝

X /�

x!H i.!˝

X /x


is surjective for everyi . But it is also injective by Theorem 5.1. This provesthatı W .!˝

X/x!RHomX .˝

0X; !˝

X/x is a quasi-isomorphism. Finally, applying

the functorRHomOX;x. ; .!˝

X/x/ one more time proves thatX is Du Bois at

x, contradicting our choice ofx 2X ˜

This also gives the following Boutot-like theorem for Du Bois singularities (cf.[Bou87]).

COROLLARY 5.4 [Kov99, Theorem 2.3; Kol95, Theorem 12.8].Suppose thatf W Y ! X is a morphism, Y has Du Bois singularities and the natural mapOX ! Rf�OY has a left inverse in the derived category. ThenX also hasDu Bois singularities.

PROOF. Observe that the composition is an isomorphism

OX !˝0X !Rf�˝

0Y 'Rf�OY !OX :

Then apply Theorem 5.3. ˜

As an easy corollary, we see that rational singularities areDu Bois (which wasfirst observed in the isolated case by Steenbrink in [Ste83, Proposition 3.7]).

COROLLARY 5.5 [Kov99; Sai00].If X has rational singularities, thenX hasDu Bois singularities.

PROOF. Let � W zX ! X be a log resolution. One has the compositionOX !

˝0X!R��O zX

. SinceX has rational singularities, this composition is a quasi-isomorphism. Apply Corollary 5.4. ˜

6. Hyperresolution-free characterizations of Du Bois singularities

The definition of Du Bois singularities given via hyperresolution is relativelycomplicated (hyperresolutions themselves can be rather complicated to com-pute; see 2). In this section we state several hyperresolution free characteri-zations of Du Bois singularities. The first such characterization was given bySteenbrink in the isolated case. Another, more analytic characterization wasgiven by Ishii and improved by Watanabe in the isolated quasi-Gorenstein3

case. Finally the second named author gave a characterization that works forany reduced scheme.

A relatively simple characterization of an affine cone over aprojective varietybeing Du Bois is given in [DB81]. Steenbrink generalized this criterion to allnormal isolated singularities. It is this criterion that Steenbrink, Ishii, Watanabe,and others used extensively to study isolated Du Bois singularities.

3A varietyX is quasi-Gorenstein ifKX is a Cartier divisor. It is not required thatX is Cohen–Macaulay.


THEOREM 6.1 [DB81, Proposition 4.13; Ste83, 3.6].Let .X;x/ be a normalisolated Du Bois singularity, and� W zX ! X a log resolution of.X;x/ suchthat� is an isomorphism outside ofX nfxg. LetE denote the reduced preimageof x. Then.X;x/ is a Du Bois singularity if and only if the natural map

Ri��O zX!Ri��OE

is an isomorphism for alli > 0.

PROOF. Using Theorem 4.2, we have an exact triangle

˝0X

// ˝0fxg˚R��˝

0zX

�// R��˝

0E

C1// :

Sincefxg, zX andE are all Du Bois (the first two are smooth, andE is snc), wehave the following exact triangle

˝0X

// Ofxg˚R��O zX�

// R��OEC1

// :

Suppose first thatX has Du Bois singularities (that is,0X'qisOX ). By taking

cohomology and examining the long exact sequence, we see that Ri��O zX!

Ri��OE is an isomorphism for alli > 0.So now suppose thatRi��O zX

!Ri��OE is an isomorphism for alli > 0.By considering the long exact sequence of cohomology, we seethatH i.˝0

X/ is

zero for alli > 0. On the other hand,H 0.˝0X/ is naturally identified with the

seminormalization ofOX ; see Proposition 7.8 below. Thus ifX is normal, thenOX !H 0.˝0

X/ is an isomorphism. ˜

We now state a more analytic characterization, due to Ishii and slightly improvedby Watanabe. First we recall the definition of the plurigenera of a singularity.

DEFINITION 6.2. For a singularity.X;x/, we define the plurigenerafımgm2N;

ım.X;x/D dimC � .X nx;OX .mKX //=L2=m.X n fxg/;

whereL2=m.X n fxg/ denotes the set of allL2=m-integrablem-uple holomor-phic n-forms onX n fxg.

THEOREM 6.3 [Ish85, Theorem 2.3; Wat87, Theorem 4.2].Letf W zX ! X bea log resolution of a normal isolated Gorenstein singularity .X;x/ of dimensionn � 2. SetE to be the reduced exceptional divisor(the preimage ofx). Then.X;x/ is a Du Bois singularity if and only ifım.X;x/� 1 for anym 2 N.

In [Sch07], a characterization of arbitrary Du Bois singularities is given that didnot rely on hyperresolutions, but instead used a single resolution of singularities.An improvement of this was also obtained in [ST08, Proposition 2.20]. Weprovide a proof for the convenience of the reader.


THEOREM 6.4 [Sch07; ST08, Proposition 2.20].LetX be a reduced separatedscheme of finite type over a field of characteristic zero. Suppose thatX � Y

whereY is smooth and suppose that� W zY ! Y is a proper birational map withzY smooth and whereX D ��1.X /red, the reduced preimage ofX , is a simplenormal crossings divisor(or in fact any scheme with Du Bois singularities).ThenX has Du Bois singularities if and only if the natural mapOX !R��OX

is a quasi-isomorphism.In fact, we can say more. There is an isomorphism

R��OX z // ˝0X

such that the natural mapOX ! ˝0X

can be identified with the natural mapOX !R��OX .

PROOF. We first assume that� is an isomorphism outside ofX . Then usingTheorem 4.2, we have an exact triangle

˝0Y

// ˝0X˚R��˝

0zY

�// R��˝

0

X

C1// :

Using the octahedral axiom, we obtain the diagram

C ˝

z››

// ˝0Y

˛››

// ˝0X

ˇ››

C1//

C ˝ // R��˝0zY

// R��˝0

X

C1// :

whereC ˝ is simply the object in the derived category that completes the trian-gles. But notice that the vertical arrowis an isomorphism sinceY has rationalsingularities (in which case each term in the middle column is isomorphic toOY ). Thus the vertical arrow is also an isomorphism.

One always has a commutative diagram

OX

››

// ˝0X

ˇ››

R��OX ı// R��˝

0

X

(where the arrows are the natural ones). Observe thatX has Du Bois singular-ities since it has normal crossings, thusı is a quasi-isomorphism. But then thetheorem is proven at least in the case that� is an isomorphism outside ofX .

For the general case, it is sufficient to show thatR��OX is independent ofthe choice of resolution. Since any two log resolutions can be dominated by a


third, it is sufficient to consider two log resolutions�1 WY1!Y and�2 WY2!Y

and a map between them� W Y2! Y1 overY . Let F1D .��11.X //red andF2D

.��12.X //redD .��1.F1//red. Dualizing the map and applying Grothendieck

duality implies that it is sufficient to prove that!Y1.F1/ R��.!Y2

.F2// is aquasi-isomorphism.

We now apply the projection formula while twisting by!�1Y1.�F1/. Thus it

is sufficient to prove that

R��.!Y2=Y1.F2� �

�F1//! OY1

is a quasi-isomorphism. But note thatF2 � ��F1 D �b�

�.1� "/F1c for suffi-ciently small"> 0. Thus it is sufficient to prove that the pair.Y1; .1�"/F1/ hasklt singularities by Kawamata–Viehweg vanishing in the form of local vanishingfor multiplier ideals; see [Laz04, 9.4]. But this is true since Y1 is smooth andF1 is a reduced integral divisor with simple normal crossings. ˜

It seems that in this characterization the condition that the ambient varietyY issmooth is asking for too much. We propose that the following may be a morenatural characterization. For some motivation and for a statement that may beviewed as a sort of converse; see Conjecture 12.5 and the discussion preceding it.

CONJECTURE6.5. Theorem6.4 should remain true if the hypothesis thatY issmooth is replaced by the condition thatY has rational singularities.

Having Du Bois singularities is a local condition, so even ifX is not embeddablein a smooth scheme, one can still use Theorem 6.4 by passing toan affine opencovering.

To illustrate the utility and meaning of Theorem 6.4, we willexplore thesituation whenX is a hypersurface inside a smooth schemeY . In the notationof Theorem 6.4, we have the diagram of exact triangles

R��O zY.�X / // R��O zY

// R��OXC1

//

0 // OY .�X /

˛

OO

// OY

ˇ

OO

// OX

OO

// 0

SinceY is smooth, is a quasi-isomorphism (as thenY has at worst rationalsingularities). Therefore,X has Du Bois singularities if and only if the mapis a quasi-isomorphism. However,˛ is a quasi-isomorphism if and only if thedual map

R��!˝

zY.X /! !˝

Y .X / (6.5.1)


is a quasi-isomorphism. The projection formula tells us that Equation 6.5.1 is aquasi-isomorphism if and only

R��O zY.K zY =Y

��X CX /!OX (6.5.2)

is a quasi-isomorphism. Note however that��X CX D d�.1� "/��X e for" > 0 and sufficiently close to zero. Thus the left side of Equation6.5.2 can beviewed asR��O zY

.dK zY =Y� .1� "/��X e/ for " > 0 sufficiently small. Note

that Kawamata–Viehweg vanishing in the form of local vanishing for multi-plier ideals implies thatJ .Y; .1� "/X /'qis R��O zY

.dK zY =Y� .1� "/��X e/.

ThereforeX has Du Bois singularities if and only ifJ .Y; .1� "/X /'OX .

COROLLARY 6.6. If X is a hypersurface in a smoothY , thenX has Du Boissingularities if and only if the pair.Y;X / is log canonical.

Du Bois hypersurfaces have also been characterized via the Bernstein–Sato poly-nomial; see [Sai09, Theorem 0.5].

7. Seminormality of Du Bois singularities

In this section we show that Du Bois singularities are partially characterizedby seminormality. First we remind the reader what it means for a scheme to beseminormal.

DEFINITION 7.1 [Swa80; GT80]. Suppose thatR is a reduced excellent ringand thatS �R is a reducedR-algebra which is finite as anR-module. We saythat the extensioni WRŒ S is subintegralif

(i) i induces a bijection on spectra, SpecS ! SpecR, and(ii) i induces an isomorphism of residue fields over every (possibly nonclosed)

point of SpecR.

REMARK 7.2. In [GT80], subintegral extensions are called quasi-isomorphisms.

DEFINITION 7.3 [Swa80; GT80]. Suppose thatR is a reduced excellent ring.We say thatR is seminormalif every subintegral extensionRŒ S is an iso-morphism. We say that a schemeX is seminormalif all of its local rings areseminormal.

REMARK 7.4. In [GT80], the authors callR seminormal if there is no propersubintegral extensionRŒ S such thatS is contained in the integral closure ofR (in its total field of fractions). However, it follows from [Swa80, Corollary3.4] that the definition above is equivalent.

REMARK 7.5. Seminormality is a local property. In particular, a ring is semi-normal if and only if it is seminormal after localization at each of its prime(equivalently, maximal) ideals.


REMARK 7.6. The easiest example of seminormal schemes are schemes withsnc singularities. In fact, a one dimensional variety over an algebraically closedfield is seminormal if and only if its singularities are locally analytically iso-morphic to a union of coordinate axes in affine space.

We will use the following well known fact about seminormality.

LEMMA 7.7. If X is a seminormal scheme andU � X is any open set, then� .U;OX / is a seminormal ring.

PROOF. We leave it as an exercise to the reader. ˜

It is relatively easy to see, using the original definition via hyperresolutions, thatif X has Du Bois singularities, then it is seminormal. Du Bois certainly knew thisfact (see [DB81, Proposition 4.9]) although he didn’t use the word seminormal.Later Saito proved that seminormality in fact partially characterizes Du Boissingularities. We give a different proof of this fact, from [Sch09].

PROPOSITION7.8 [Sai00, Proposition 5.2; Sch09, Lemma 5.6].Suppose thatX is a reduced separated scheme of finite type overC. Thenh0.˝0

X/ D OX sn

whereOX sn is the structure sheaf of the seminormalization ofX .

PROOF. Without loss of generality we may assume thatX is affine. We needonly consider��OE by Theorem 6.4. By Lemma 7.7,��OE is a sheaf ofseminormal rings. Now letX 0 D Spec.��OE/ and consider the factorization

E! X 0!X:

Note E ! X 0 must be surjective since it is dominant by construction and isproper by [Har77, II.4.8(e)]. Since the composition has connected fibers, somust have� W X 0! X . On the other hand,� is a finite map since� is proper.Therefore� is a bijection on points. Because these maps and schemes are offinite type over an algebraically closed field of characteristic zero, we see that� .X;OX /! � .X 0;OX 0/ is a subintegral extension of rings. SinceX 0 is semi-normal, so is� .X 0;OX 0/, which completes the proof. ˜

8. A multiplier-ideal-like characterization of Cohen–MacaulayDu Bois singularities

In this section we state a characterization of Cohen–Macaulay Du Bois singu-larities that explains why Du Bois singularities are so closely linked to rationaland log canonical singularities.

We first do a suggestive computation. Suppose thatX embeds into a smoothschemeY and that� W zY ! Y is an embedded resolution ofX in Y that is anisomorphism outside ofX . Set zX to be the strict transform ofX and setX to


be the reduced preimage ofX . We further assume thatX D zX [E whereE

is a reduced simple normal crossings divisor that intersects zX transversally inanother reduced simple normal crossing divisor. Note thatE is the exceptionaldivisor of � (with reduced scheme structure). Set˙ � X be the image ofE.We have the short exact sequence

0!O zX.�E/! OX !OE! 0

We applyRHomOY. ; !˝

zY/ followed byR�� and obtain the exact triangle

R��!˝

E// R��!

˝

X// R��! zX

.E/ŒdimX �C1

//

Using condition (i) in Theorem 4.2, the leftmost object can be identified withRHomO˙

.˝0˙; !˝

˙/ and the middle object,R��!

˝

X, can be identified with

RHomOX.˝0

X; !˝

X/. Recall thatX has Du Bois singularities if and only if the

natural mapRHomOX.˝0

X; !˝

X/!!˝

Xis an isomorphism. Therefore, the object

��! zX.E/ is closely related to whether or notX has Du Bois singularities. This

inspired the following result, which we do not prove.

THEOREM 8.1 [KSS10, Theorem 3.1] .Suppose thatX is normal and Cohen–Macaulay. Let � W X 0 ! X be a log resolution, and denote the reduced ex-ceptional divisor of� by G. ThenX has Du Bois singularities if and only if��!X 0.G/' !X .

We mention that the main idea in the proof is to show that

��!X 0.G/'H � dimX�

RHomOX.˝0

X ; !˝

X /�

:

Related results can also be obtained in the nonnormal Cohen–Macaulay case;see [KSS10] for details.

REMARK 8.2. The submodule��!X 0.G/ � !X is independent of the choiceof log resolution. Thus this submodule may be viewed as an invariant whichpartially measures how far a scheme is from being Du Bois (compare with[Fuj08]).

As an easy corollary, we obtain another proof that rational singularities areDu Bois (this time via the Kempf-criterion for rational singularities).

COROLLARY 8.3. If X has rational singularities, thenX has Du Bois singu-larities.

PROOF. SinceX has rational singularities, it is Cohen–Macaulay and normal.Then��!X 0 D !X but we also have��!X 0 � ��!X 0.G/ � !X , and thus��!X 0.G/D !X as well. Then use Theorem 8.1. ˜

We also see immediately that log canonical singularities coincide with Du Boissingularities in the Gorenstein case.


COROLLARY 8.4. Suppose thatX is Gorenstein and normal. Then X isDu Bois if and only ifX is log canonical.

PROOF. X is easily seen to be log canonical if and only if��!X 0=X .G/'OX .The projection formula then completes the proof. ˜

In fact, a slightly improved version of this argument can be used to show thatevery Cohen–Macaulay log canonical pair is Du Bois; see [KSS10, Theorem3.16].

9. The Kollar–Kovacs splitting criterion

The proof of the following, rather flexible, criterion for DuBois singularitiescan be found in the original paper.

THEOREM 9.1 [KK10]. Let f W Y ! X be a proper morphism between re-duced schemes of finite type overC, W � X an arbitrary subscheme, andF WD f �1.W /, equipped with the induced reduced subscheme structure. LetIW �X denote the ideal sheaf ofW in X andIF�Y the ideal sheaf ofF in Y .Assume that the natural map%

IW �X %// Rf�IF�Y

%0

{{

O

W_

g

o

admits a left inverse%0, that is, �0 ı �D idIW �X. Then ifY;F , andW all have

DB singularities, so doesX .

REMARK 9.1.1. Notice that it is not required thatf be birational. On the otherhand the assumptions of the theorem and [Kov00a, Theorem 1] imply that ifY nF has rational singularities, e.g., ifY is smooth, thenX nW has rationalsingularities as well.

This theorem is used to derive various consequences in [KK10], some of whichare formally unrelated to Du Bois singularities. We will mention some of thesein the sequel, but the interested reader should look at the original article to obtainthe full picture.

10. Log canonical singularities are Du Bois

Log canonical and Du Bois singularities are very closely related as we haveseen in the previous sections. This was first observed in [Ish85]; see also [Wat87]and [Ish87].

Recently, Kollar and the first named author gave a proof that log canonicalsingularities are Du Bois using Theorem 9.1. We will sketch some ideas of the


proof here. There are two main steps. First, one shows that the non-klt locusof a log canonical singularity is Du Bois (this generalizes [Amb98] and [Sch08,Corollary 7.3]). Then one uses Theorem 9.1 to show that this property is enoughto conclude thatX itself is Du Bois. For the first part we refer the reader to theoriginal paper. The key point of the second part is containedin the followingLemma. Here we give a different proof than in [KK10].

LEMMA 10.1.Suppose.X; �/ is a log canonical pair and that the reduced non-klt locus of.X; �/ has Du Bois singularities. ThenX has Du Bois singularities.

PROOF. First recall that the multiplier idealJ .X; �/ is precisely the definingideal of the non-klt locus of.X; �/ and since.X; �/ is log canonical, it is aradical ideal. We set �X to be the reduced subscheme ofX defined by thisideal. Since the statement is local, we may assume thatX is affine and thus thatX is embedded in a smooth schemeY . We let� W zY ! Y be an embeddedresolution of.X; �/ in Y and we assume that� is an isomorphism outside thesingular locus ofX . Set˙ to be the reduced-preimage of (which we mayassume is a divisor inzY ) and let zX denote the strict transform ofX . We considerthe diagram of exact triangles

A˝

˛

››

// B˝

ˇ

››

// C ˝

››

C1//

0 // J .X; �/

ı››

// OX

››

// O˙

"

››

// 0

R��O zX.�˙/ // R��O

˙[ zX// R��O˙

C1//

Here the first row is made up of objects inDbcoh.X / needed to make the columns

into exact triangles. Since has Du Bois singularities, the map" is an isomor-phism and soC ˝'0. On the other hand, there is a natural mapR��O zX

.�˙/!

R��O zX.K zX

��.KX C�//' J .X; �/ since.X; �/ is log canonical. Thisimplies that the map is the zero map in the derived category. However,we then see that is also zero in the derived category which implies thatOX ! R��O

˙[ zXhas a left inverse. Therefore,X has Du Bois singularities

(since˙[ zX has simple normal crossing singularities) by Theorems 5.3 and 6.4.˜

11. Applications to moduli spaces and vanishing theorems

The connection between log canonical and Du Bois singularities have manyuseful applications in moduli theory. We will list a few without proof.


SETUP 11.1. Let� WX !B be a flat projective morphism of complex varietieswith B connected. Assume that for allb 2B there exists aQ-divisor Db onXb

such that.Xb;Db/ is log canonical.

REMARK 11.2. Notice that it is not required that the divisorsDb form a family.

THEOREM 11.3 [KK10]. Under the assumptions11.1,hi.Xb;OXb/ is indepen-

dent ofb 2B for all i .

THEOREM 11.4 [KK10]. Under the assumptions11.1, if one fiber of� isCohen–Macaulay(resp. Sk for somek), so are all the fibers.

THEOREM 11.5 [KK10]. Under the assumptions11.1,the cohomology sheaveshi.!˝

�/ are flat overB, where!˝

�denotes the relative dualizing complex of�.

Du Bois singularities also appear naturally in vanishing theorems. As a cul-mination of the work of Tankeev, Ramanujam, Miyaoka, Kawamata, Viehweg,Koll ar, and Esnault–Viehweg, Kollar proved a rather general form of a Kodaira-type vanishing theorem in [Kol95, 9.12]. Using the same ideas this was slightlygeneralized to the following theorem in [KSS10].

THEOREM 11.6 [Kol95, 9.12; KSS08, 6.2].LetX be a proper variety andL aline bundle onX . LetL m'OX .D/, whereDD

P

diDi is an effective divisor,and lets be a global section whose zero divisor isD. Assume that0 < di < m

for everyi . Let Z be the scheme obtained by taking them-th root ofs (that is,Z DX Œ

ps� using the notation from[Kol95, 9.4]). Assume further that

H j .Z;CZ /!H j .Z;OZ /

is surjective. Then, for any collection ofbi � 0, the natural map

H j�

X;L �1�

�P

biDi

��

!H j .X;L �1/

is surjective.

This, combined with the fact that log canonical singularities are Du Bois, yieldsthat Kodaira vanishing holds for log canonical pairs:

THEOREM 11.7 [KSS10, 6.6].Kodaira vanishing holds for Cohen–Macaulaysemi-log canonical varieties: Let.X; �/ be a projective Cohen–Macaulay semi-log canonical pair andL an ample line bundle onX . ThenH i.X;L �1/D 0

for i < dimX .

It turns out that Du Bois singularities appear naturally in other kinds of vanishingtheorems. We cite one here.


THEOREM 11.8 [GKKP10, 9.3].Let .X;D/ be a log canonical reduced pairof dimensionn� 2, � W zX ! X a log resolution with�-exceptional setE, andzD D Supp

�

EC��1D�

. Then

Rn�1��O zX.� zD/D 0:

12. Deformations of Du Bois singularities

Given the importance of Du Bois singularities in moduli theory it is an im-portant obvious question whether they are invariant under small deformation.

It is relatively easy to see from the construction of the Du Bois complexthat a general hyperplane section (or more generally, the general member of abase point free linear system) on a variety with Du Bois singularities again hasDu Bois singularities. Therefore the question of deformation follows from thefollowing.

CONJECTURE12.1. (cf . [Ste83]) Let D � X be a reduced Cartier divisorand assume thatD has only Du Bois singularities in a neighborhood of a pointx 2 D. ThenX has only Du Bois singularities in a neighborhood of the pointx.

This conjecture was proved for isolated Gorenstein singularities by Ishii [Ish86].Also note that rational singularities satisfy this property; see [Elk78].

We also have the following easy corollary of the results presented earlier:

THEOREM 12.2. Assume thatX is Gorenstein andD is normal.4 Then thestatement of Conjecture12.1is true.

PROOF. The question is local so we may restrict to a neighborhood ofx. If X

is Gorenstein, then so isD as it is a Cartier divisor. ThenD is log canonical by(8.4), and then the pair.X;D/ is also log canonical by inversion of adjunction[Kaw07]. (Recall that ifD is normal, then so isX alongD). This implies thatX is also log canonical and thus Du Bois. ˜

It is also stated in [Kov00b, 3.2] that the conjecture holds in full generality.Unfortunately, the proof is not complete. The proof published there works ifone assumes that the non-Du Bois locus ofX is contained inD. For instance,one may assume that this is the case if the non-Du Bois locus isisolated.

The problem with the proof is the following: it is stated thatby taking hyper-plane sections one may assume that the non-Du Bois locus is isolated. However,this is incorrect. One may only assume that theintersectionof the non-Du Boislocus ofX with D is isolated. If one takes a further general section then it will

4This condition is actually not necessary, but the proof becomes rather involved without it.


miss the intersection point and then it is not possible to make any conclusionsabout that case.

Therefore currently the best known result with regard to this conjecture is thefollowing:

THEOREM 12.3 [Kov00b, 3.2].Let D � X be a reduced Cartier divisor andassume thatD has only Du Bois singularities in a neighborhood of a pointx 2D and thatX nD has only Du Bois singularities. ThenX has only Du Boissingularities in a neighborhood ofx.

Experience shows that divisors not in general position tendto have worse sin-gularities than the ambient space in which they reside. Therefore one would infact expect that ifX nD is reasonably nice, andD has Du Bois singularities,then perhapsX has even better ones.

We have also seen that rational singularities are Du Bois andat least Cohen–Macaulay Du Bois singularities are not so far from being rational cf. 8.1. Thefollowing result of the second named author supports this philosophical point.

THEOREM 12.4 [Sch07, Theorem 5.1].Let X be a reduced scheme of finitetype over a field of characteristic zero, D a Cartier divisor that has Du Boissingularities and assume thatXnD is smooth. ThenX has rational singularities(in particular, it is Cohen–Macaulay).

Let us conclude with a conjectural generalization of this statement:

CONJECTURE12.5. Let X be a reduced scheme of finite type over a field ofcharacteristic zero, D a Cartier divisor that has Du Bois singularities and as-sume thatX nD has rational singularities. ThenX has rational singularities(in particular, it is Cohen–Macaulay).

Essentially the same proof as in (12.2) shows that this is also true under thesame additional hypotheses.

THEOREM 12.6. Assume thatX is Gorenstein andD is normal.5 Then thestatement of Conjecture12.5is true.

PROOF. If X is Gorenstein, then so isD as it is a Cartier divisor. Then by (8.4)D is log canonical. Then by inversion of adjunction [Kaw07] the pair.X;D/ isalso log canonical nearD. (Recall that ifD is normal, then so isX alongD).

As X is Gorenstein andX nD has rational singularities, it follows thatX nD

has canonical singularities. ThenX has only canonical singularities everywhere.This can be seen by observing thatD is a Cartier divisor and examining thediscrepancies that lie overD for .X;D/ as well as forX . Therefore, by [Elk81],X has only rational singularities alongD. ˜

5Again, this condition is not necessary, but makes the proof simpler.


13. Analogs of Du Bois singularities in characteristicp > 0

Starting in the early 1980s, the connections between singularities defined bythe action of the Frobenius morphism in characteristicp > 0 and singularitiesdefined by resolutions of singularities started to be investigated, cf. [Fed83].After the introduction of tight closure in [HH90], a precisecorrespondence be-tween several classes of singularities was established. See, for example, [FW89;MS91; HW02; Smi97; Har98; MS97; Smi00; Har05; HY03; Tak04; TW04;Tak08]. The second named author partially extended this correspondence in hisdoctoral dissertation by linking Du Bois singularities with F -injective singular-ities, a class of singularities defined in [Fed83]. The currently known implica-tions are summarized below.

log terminalrz $,

+3

¸«

rational

¸«

rz $,

F -regular +3

¸«

F -rational

¸«

log canonical +3

dlV^

+ Gor. and normal

Du Boisem

F -Pure/F -split +3

X‘

+ Gor.

F -injective

We will give a short proof that normal Cohen–Macaulay singularities of denseF -injective type are Du Bois, based on the characterization of Du Bois singu-larities given in Section 8.

Note that Du Bois andF -injective singularities also share many commonproperties. For exampleF -injective singularities are also seminormal [Sch09,Theorem 4.7].

First however, we will defineF -injective singularities (as well as some nec-essary prerequisites).

DEFINITION 13.1. Suppose thatX is a scheme of characteristicp > 0 withabsolute Frobenius mapF W X ! X . We say thatX is F -finite if F�OX isa coherentOX -module. A ringR is calledF -finite if the associated schemeSpecR is F -finite.

REMARK 13.2. Any scheme of finite type over a perfect field isF -finite; seefor example [Fed83].

DEFINITION 13.3. Suppose that.R;m/ is anF -finite local ring. We say thatRis F -injective if the induced Frobenius mapF W H i

m.R/! H i

m.R/ is injective

for everyi > 0. We say that anF -finite scheme isF -injectiveif all of its stalksareF -injective local rings.


REMARK 13.4. If .R;m/ is F -finite, F -injective and has a dualizing complex,thenRQ is alsoF -injective for anyQ2SpecR. This follows from local duality;see [Sch09, Proposition 4.3] for details.

LEMMA 13.5. SupposeX is a Cohen–Macaulay scheme of finite type over aperfect fieldk. ThenX is F -injective if and only if the natural mapF�!X !!X

is surjective.

PROOF. Without loss of generality (sinceX is Cohen–Macaulay) we can assumethatX is equidimensional. Setf W X ! Speck to be the structural morphism.SinceX is finite type over a perfect field, it has a dualizing complex!˝

XD f !k

and we set!X D h� dimX .!˝

X/. SinceX is Cohen–Macaulay,X is F -injective

if and only if the Frobenius mapH dimXx .OX ;x/

// H dimXx .F�OX ;x/ is in-

jective for every closed pointx 2 X . By local duality (see [Har66, Theorem6.2] or [BH93, Section 3.5]) such a map is injective if and only if the dual mapF�!X ;x! !X ;x is surjective. But that map is surjective, if and only if the mapof sheavesF�!X ! !X is surjective. ˜

We now briefly describe reduction to characteristicp > 0. Excellent and farmore complete references include [HH09, Section 2.1] and [Kol96, II.5.10].Also see [Smi01] for a more elementary introduction.

Let R be a finitely generated algebra over a fieldk of characteristic zero.Write R D kŒx1; : : : ;xn�=I for some idealI and letS denotekŒx1; : : : ;xn�.Let X D SpecR and� W zX ! X a log resolution ofX corresponding to theblow-up of an idealJ . Let E denote the reduced exceptional divisor of�. ThenE is the subscheme defined by the radical of the idealJ �O zX

.There exists a finitely generatedZ-algebraA � k that includes all the coef-

ficients of the generators ofI andJ , a finitely generatedA algebraRA � R,an idealJA � RA, and schemeszXA and EA of finite type overA such thatRA˝A k DR, JARD J , zXA �SpecA Speck DX andEA �SpecA Speck DE

with EA an effective divisor with support defined by the idealJA � O zXA. We

may localizeA at a single element so thatYA is smooth overA andEA is areduced simple normal crossings divisor overA. By further localizingA (at asingle element), we may assume any finite set of finitely generatedRA modulesis A-free (see [Hun96, 3.4] or [HR76, 2.3]) and we may assume thatA itself isregular. We may also assume that a fixed affine cover ofEA and a fixed affinecover of zXA are alsoA-free.

We will now form a family of positive characteristic models of X by lookingat all the ringsRt D RA˝A k.t/ wherek.t/ is the residue field of a maximalideal t 2 T D SpecA. Note thatk.t/ is a finite, and thus perfect, field ofcharacteristicp. We may also tensor the various schemesXA, EA, etc. withk.t/ to produce a characteristicp model of an entire situation.


By making various cokernels of maps freeA-modules, we may also assumethat maps between modules that are surjective (respectively injective) overkcorrespond to surjective (respectively injective) maps overA, and thus surjective(respectively injective) in our characteristicp model as well; see [HH09] fordetails.

DEFINITION 13.6. A ring R of characteristic zero is said to have denseF -injective type if for every family of characteristicp� 0 models withA chosensufficiently large, a Zariski dense set of those models (overSpecA) haveF -injective singularities.

THEOREM 13.7 [Sch09].Let X be a reduced scheme of finite type overC andassume that it has denseF -injective type. ThenX has Du Bois singularities.

PROOF. We only provide a proof in the case thatX is normal and Cohen–Macaulay. For a complete proof, see [Sch09]. Let� W zX!X be a log resolutionof X with exceptional divisorE. We reduce this entire setup to characteristicp� 0 such that the correspondingX is F -injective. LetF e W X ! X be thee-iterated Frobenius map.

We have the commutative diagram

F e��! zX

.peE/

�

››

// ��! zX.E/

ˇ

››

F e�!X

�// !X

where the horizontal arrows are induced by the dual of the Frobenius map,OX ! F e

�OX , and the vertical arrows are the natural maps induced by�. Byhypothesis,� is surjective. On the other hand, fore > 0 sufficiently large, themap labeled� is an isomorphism. Therefore the map� ı � is surjective whichimplies that the map is also surjective. But as this holds for a dense set ofprimes, it must be surjective in characteristic zero as well, and in particular, asa consequenceX has Du Bois singularities. ˜

It is not known whether the converse of this statement is true:

OPEN PROBLEM 13.8. If X has Du Bois singularities, does it have dense F-injective type?

SinceF -injective singularities are known to be closely related toDu Bois sin-gularities, it is also natural to ask howF -injective singularities deform cf. Con-jecture 12.1. In general, this problem is also open.

OPENPROBLEM 13.9. If a Cartier divisorD in X hasF -injective singularities,doesX haveF -injective singularities nearD?


In the case thatX (equivalentlyD) is Cohen–Macaulay, the answer is affirma-tive, see [Fed83]. In fact, Fedder definedF -injective singularities partly becausethey seemed to deform better thanF -pure singularities (the conjectured analogof log canonical singularities).

Appendix A. Connections with Buchsbaum rings

In this section we discuss the links between Du Bois singularities and Buchs-baum rings. Du Bois singularities are not necessarily Cohen–Macaulay, but inmany cases, they are Buchsbaum (a weakening of Cohen–Macaulay).

Recall that a local ring.R;m; k/ hasquasi-Buchsbaumsingularities if

mH im.R/D 0

for all i<dimR. Further recall that a ring is calledBuchsbaumif �dimRR�m.R/

is quasi-isomorphic to a complex ofk-vector spaces. Here�dimR is the brutaltruncation of the complex at the dimR location. Note that this is not the usualdefinition of Buchsbaum singularities, rather it is the so-called Schenzel’s cri-terion; see [Sch82]. Notice that Cohen–Macaulay singularities are Buchsbaum(after truncation, one obtains the zero-object in the derived category).

It was proved by Tomari that isolated Du Bois singularities are quasi-Buchs-baum (a proof can be found in [Ish85, Proposition 1.9]), and then by Ishida thatisolated Du Bois singularities were in fact Buchsbaum. Herewe briefly reviewthe argument to show that isolated Du Bois singularities arequasi-Buchsbaumsince this statement is substantially easier.

PROPOSITIONA.1. Suppose that.X;x/ is an isolated Du Bois singularity withRD OX ;x . ThenR is quasi-Buchsbaum.

PROOF. Note that we may assume thatX is affine. Since SpecR is regularoutside its the maximal idealm, it is clear that some power ofm annihilatesH i

m.R/ for all i < dimR. We need to show that the smallest power for which

this happens is1. We let� W zX!X be a log resolution with exceptional divisorE as in Theorem 6.1. SinceX is affine, we see thatH i

m.R/ ' H i�1.X n

fmg;OX / ' H i�1. zX nE;O zX/ for all i > 0. Therefore, it is enough to show

that mH i�1. zX nE;O zX/ D 0 for all i < dimX . In other words, we need to

show thatmH i. zX nE;O zX/D 0 for all i < dimX � 1.

We examine the long exact sequence

: : : // H i�1. zX nE;O zX/ // H i

E. zX ;O zX

/ // H i. zX ;O zX/

// H i. zX nE;O zX/ // : : :


Now, H iE. zX ;O zX

/ D Ri.�m ı ��/ .OX / which vanishes fori < dimX by theMatlis dual of Grauert-Riemenschneider vanishing. Therefore

H i. zX nE;O zX/'H i. zX ;O zX

/

for i < dimX � 1. Finally, sinceX is Du Bois,H i. zX ;O zX/ D H i.E;OE/

by Theorem 6.1. But it is obvious thatmH i.E;OE/D 0 sinceE is a reduceddivisor whose image inX is the point corresponding tom. The result thenfollows. ˜

It is easy to see that isolatedF -injective singularities are also quasi-Buchsbaum.

PROPOSITION A.2. Suppose that.R;m/ is a local ring that isF -injective.Further suppose thatSpecR n fmg is Cohen–Macaulay. Then.R;m/ is quasi-Buchsbaum.

PROOF. Since the punctured spectrum ofR is Cohen–Macaulay,H im.R/ is

annihilated by some power ofm for i < dimR. We will show that the smallestsuch power is1. Choosec 2m. SinceR is F -injective,F e WH i

m.R/!H i

m.R/

is injective for alle> 0. Choosee large enough so thatcpe

H im.R/ is zero for all

i < e. However, for any elementz 2H im.R/, F e.cz/D cpe

F e.z/2 cpe

H im.R/D

0 for i <dimR. This implies thatczD 0 and somH im.R/D 0 for i <dimR. ˜

Perhaps the most interesting open question in this area is the following:

OPEN PROBLEM 1.3 (TAKAGI ). Are F -injective singularities with isolatednon-CM locus Buchsbaum?

Given the close connection betweenF -injective and Du Bois singularities, thisquestion naturally leads to the next one:

OPEN PROBLEM 1.4. Are Du Bois singularities with isolated non-CM locusBuchsbaum?

2. Cubical hyperresolutions

For the convenience of the reader we include a short appendixexplainingthe construction of cubical hyperresolutions, as well as several examples. Wefollow [GNPP88] and mostly use their notation.

First let us fix a small universe to work in. Let Schdenote the category ofreduced schemes. Note that the usual fibred product of schemesX�S Y need notbe reduced, even whenX andY are reduced. We wish to construct the fibredproduct in the category of reduced schemes. Given any schemeW (reducedor not) with maps toX and Y over S , there is always a unique morphismW !X �S Y , which induces a natural unique morphismWred! .X �S Y /red.


It is easy to see that.X �S Y /red is the fibred product in the category of reducedschemes.

Let us denote by1 the categoryf0g and by2 the categoryf0 ! 1g. Letn be an integer� �1. We denote by C

n the product ofnC 1 copies of thecategory2D f0! 1g [GNPP88, I, 1.15]. The objects ofCn are identified withthe sequences D .˛0; ˛1; : : : ; ˛n/ such that i 2 f0; 1g for 0 � i � n. FornD�1, we set C

�1Df0g and fornD 0 we have C

0Df0! 1g. We denote by

˜n the full subcategory consisting of all objects of˜Cn except the initial object

.0; : : : ; 0/. Clearly, the category Cn can be identified with the category ofn

with an augmentation map tof0g.

DEFINITION 2.1. A diagram of schemesis a functor˚ from a categoryCop tothe category of schemes. Afinite diagram of schemesis a diagram of schemessuch that the aforementioned categoryC has finitely many objects and mor-phisms; in this case such a functor will be called aC-scheme. A morphism ofdiagrams of schemes W Cop! Schto W Dop! Schis the combined dataof a functor� W Cop! Dop together with a natural transformation of functors� W ˚ ! ı� .

REMARK 2.2. With these definitions, the class of (finite) diagrams ofschemescan be made into a category. Likewise the set ofC-schemes can also be madeinto a category (where the functor� W Cop! Cop is always chosen to be theidentity functor).

REMARK 2.3. LetI be a category. If instead of a functor to the category of re-duced schemes, one considers a functor to the category of topological spaces, orthe category of categories, one can defineI -topological spaces, andI -categoriesin the obvious way.

If X˝W Iop!Schis anI -scheme, andi 2ObI , thenXi will denote the scheme

corresponding toi . Likewise if� 2Mor I is a morphism� W j! i , thenX� willdenote the corresponding morphismX� WXi!Xj . If f WY

˝!X

˝is a morphism

of I -schemes, we denote byfi the induced morphismYi ! Xi . If X˝is anI -

scheme, a closed sub-I -scheme is a morphism ofI -schemesg WZ˝! X

˝such

that for eachi 2 I , the mapgi WZi!Xi is a closed immersion. We will oftensuppress theg of the notation if no confusion is likely to arise. More generally,any property of a morphism of schemes (projective, proper, separated, closedimmersion, etc...) can be generalized to the notion of a morphism ofI -schemesby requiring that for each objecti of I , gi has the desired property (projective,proper, separated, closed immersion, etc...)

DEFINITION 2.4 [GNPP88, I, 2.2]. Suppose thatf W Y˝!X

˝is a morphism of

I -schemes. Define thediscriminant off to be the smallest closed sub-I -scheme


Z˝of X

˝such thatfi W .Yi� .f

�1i .Zi///! .Xi�Zi/ is an isomorphism for all

i .

DEFINITION 2.5 [GNPP88, I, 2.5]. LetS˝

be anI -scheme,f W X˝! S

˝a

proper morphism ofI -schemes, andD˝the discriminant off . We say thatf is

a resolution6 of S˝if X

˝is a smoothI -scheme (meaning that eachXi is smooth)

and dimf �1i .Di/ < dimSi , for all i 2ObI .

REMARK 2.6. This is the definition found in [GNPP88]. Note that the mapsare not required to be surjective (of course, the ones one constructs in practiceare usually surjective).

Consider the following example: the mapkŒx;y�=.xy/! kŒx� which sendsy to 0. We claim that the associated map of schemes is a “resolution” of the�-scheme SpeckŒx;y�=.xy/. The discriminant is SpeckŒx;y�=.x/. However,the preimage is simply the origin onkŒx�, which has lower dimension than “1”.Resolutions like this one are sometimes convenient to consider.

On the other hand, this definition seems to allow something itperhaps shouldnot. Choose any varietyX of dimension greater than zero and a closed pointz2X . Consider the mapz!X and consider the�-schemeX . The discriminantis all of X . However, the preimage ofX is still just a point, which has lowerdimension thanX itself, by hypothesis.

In view of these remarks, sometimes it is convenient to assume also thatdimDi < dimSi for eachi 2 ObI . In the resolutions ofI -schemes that weconstruct (in particular, in the ones that are used to that prove cubic hyperreso-lutions exist), this always happens.

Let I be a category. The set of objects ofI are given the preorder relationdefined byi � j if and only if HomI .i; j / is nonempty. We will say that acategoryI is ordered if this preorder is a partial order and, for eachi 2 ObI ,the only endomorphism ofi is the identity [GNPP88, I, C, 1.9]. Note that acategoryI is ordered if and only if all isomorphisms and endomorphismsof I

are the identity.It turns out of that resolutions ofI -schemes always exist under reasonable

hypotheses.

THEOREM 2.7 [GNPP88, I, Theorem 2.6].Let S be anI -scheme of finite typeover a fieldk. Suppose thatk is a field of characteristic zero and thatI is afinite ordered category. Then there exists a resolution ofS .

In order to construct a resolutionY˝of anI -schemeX

˝, it might be tempting to

simply resolve eachXi , setYi equal to that resolution, and somehow combine

6A resolution is a distinct notion from a cubic hyperresolution.


this data together. Unfortunately this cannot work, as shown by the examplebelow.

EXAMPLE 2.8. Consider the pinch point singularity,

X D SpeckŒx;y; z�=.x2y � z2/D SpeckŒs; t2; st �;

and letZ be the closed subscheme defined by the ideal.s; st/ (this is the singularset). LetI be the categoryf0! 1g. Consider theI -scheme defined byX0DX

andX1 D Z (with the closed immersion as the map).X1 is already smooth,and if one resolvesX0, (that is, normalizes it) there is no compatible way tomapX1 (or even another birational model ofX1) to it, since its preimage bynormalization will be two-to-one ontoZ�X ! The way this problem is resolvedis by creating additional components. So to construct a resolution Y

˝we set

Y1DZDX1 (since it was already smooth) and setY0DX 0

`

Z whereX 0 isthe normalization ofX0. The mapY1!Y0 just sendsY1 (isomorphically) to thenew component and the mapY0!X0 is the disjoint union of the normalizationand inclusion maps.

One should note that although the theorem proving the existence of resolutionsof I -schemes is constructive, [GNPP88], it is often easier in practice to constructan ad-hoc resolution.

Now that we have resolutions ofI -schemes, we can discuss cubic hyperres-olutions of schemes, in fact, even diagrams of schemes have cubic hyperresolu-tions! First we will discuss a single iterative step in the process of constructingcubic hyperresolutions. This step is called a2-resolution.

DEFINITION 2.9 [GNPP88, I, 2.7]. LetS be anI -scheme andZ˝

a ˜C

1� I -

scheme. We say thatZ˝is a2-resolutionof S if Z

˝is defined by the following

Cartesian square (pullback, or fibred product in the category of (reduced)I -schemes) of morphisms ofI -schemes:

Z11ffl

�

//

››

Z01

f

››

Z10ffl

�

// Z00

Here

(i) Z00 D S ,(ii) Z01 is a smoothI -scheme,(iii) The horizontal arrows are closed immersions ofI -schemes,(iv) f is a properI -morphism, and(v) Z10 contains the discriminant off ; in other words,f induces an isomor-

phism of.Z01/i � .Z11/i over.Z00/i � .Z10/i , for all i 2ObI .


Clearly 2-resolutions always exist under the same hypotheses that resolutionsof I -schemes exist: setZ01 to be a resolution,Z10 to be discriminant (or anyappropriate proper closed sub-I -scheme that contains it), andZ11 its (reduced)preimage inZ01.

EXAMPLE 2.10. LetI D f0g and letS be theI -scheme SpeckŒt2; t3�. LetZ01 DA1 D SpeckŒt � andZ01! S DZ00 be the map defined bykŒt2; t3�!

kŒt �. The discriminant of that map is the closed subscheme ofS DZ00 definedby the map� W kŒt2; t3�! k that sendst2 and t3 to zero. Finally we need todefineZ11. The usual fibered product in the category of schemes iskŒt �=.t2/,but we work in the category of reduced schemes, so instead thefibered productis simply the associated reduced scheme (in this case SpeckŒt �=.t/). Thus our2-resolution is defined by this diagram of rings:

kŒt �=.t/

kŒt �=.t/

99

s

s

s

s

s

s

s

s

s

s

kŒt �

ccH

H

H

H

H

H

H

H

H

kŒt2; t3�

;;

w

w

w

w

w

w

w

w

w

eeK

K

K

K

K

K

K

K

K

We need one more definition before defining a cubic hyperresolution,

DEFINITION 2.11 [GNPP88, I, 2.11]. Letr be an integer greater than or equalto 1, and letX n

˝

be a˜Cn � I -scheme, for1 � n � r . Suppose that for alln,

1� n� r , the˜C

n�1� I -schemesX nC1

00˝

andX n1˝

are equal. Then we define, byinduction onr , a ˜

Cr � I -scheme

Z˝D red.X 1

˝

;X 2˝

; : : : ;X r˝

/

that we call thereductionof .X 1˝

; : : : ;X r˝

/, in the following way: Ifr D 1, onedefinesZ

˝DX 1

˝

, if r D 2 one definesZ˝˝D red.X 1

˝

;X 2˝

/ by

Z˛ˇ D

(

X 10ˇ

if ˛ D .0; 0/;

X 2˛ˇ

if ˛ 2˜1;

for all ˇ 2˜C

0, with the obvious morphisms. Ifr > 2, one definesZ

˝recursively

as red.red.X 1˝

; : : : ;X r�1˝

/;X r˝

/.

Finally we are ready to define cubic hyperresolutions.

DEFINITION 2.12 [GNPP88, I, 2.12]. LetS be anI -scheme. Acubic hyper-resolution augmented overS is a˜

Cr � I -schemeZ

˝such that

Z˝D red.X 1

˝

; : : : ;X r˝

/;


whereX 1˝

is a2-resolution ofS , X nC1˝

is a 2-resolution ofX n1

for 1 � n < r ,andZ˛ is smooth for all 2˜r .

Now that we have defined cubic hyperresolutions, we should note that they existunder reasonable hypotheses:

THEOREM 2.13 [GNPP88, I, 2.15].Let S be anI -scheme. Suppose thatk isa field of characteristic zero and thatI is a finite(bounded) ordered category.Then there existsZ

˝, a cubic hyperresolution augmented overS such that

dimZ˛ � dimS � j˛jC 1 for all ˛ 2˜r :

Below are some examples of cubic hyperresolutions.

EXAMPLE 2.14. Let us begin by computing cubic hyperresolutions of curvesso letC be a curve. We begin by taking a resolution� W C ! C (whereC

is just the normalization). LetP be the set of singular points ofC ; thusP isthe discriminant of�. Finally we letE be the reduced exceptional set of�,therefore we have the Cartesian square

E //

››

C

�

››

P // C

It is clearly already a2-resolution ofC and thus a cubic-hyperresolution ofC .

EXAMPLE 2.15. Let us now compute a cubic hyperresolution of a schemeX

whose singular locus is itself a smooth scheme, and whose reduced exceptionalset of a strong resolution� W zX ! X is smooth (for example, any cone over asmooth variety). As in the previous example, let˙ be the singular locus ofXandE the reduced exceptional set of�, Then the Cartesian square of reducedschemes

E //

››

zX

�

››

˙ // X

is in fact a2-resolution ofX , just as in the case of curves above.

The obvious algorithm used to construct cubic hyperresolutions does not con-struct hyperresolutions in the most efficient or convenientway possible. Forexample, applying the obvious algorithm to the intersection of three coordinateplanes gives us the following.

EXAMPLE 2.16. LetX [ Y [ Z be the three coordinate planes inA3. Inthis example we construct a cubic hyperresolution using theobvious algorithm.


What makes this construction different, is that the dimension is forced to dropwhen forming the discriminant of a resolution of a diagram ofschemes.

Again we begin the algorithm by taking a resolution and the obvious one is� W .X tY tZ/! .X [Y [Z/. The discriminant isBD .X \Y /[.X \Z/[

.Y \Z/, the three coordinate axes. The fiber product making the square belowCartesian is simply the exceptional setE shown:

E D ..X\Y /[.X\Z// t ..Y \X /[.Y \Z// t ..Z\X /[.Z\Y // //

�

››

XtY tZ

�

››

B D .X\Y / [ .X\Z/[ .Y \Z/ // X[Y [Z

We now need to take a2-resolution of the2-scheme� W E ! B. We take theobvious resolution that simply separates irreducible components. This gives uszE! zB mapping to� W E! B. The discriminant ofzE! E is a set of threepointsX0, Y0 andZ0 corresponding to the origins inX , Y andZ respectively.The discriminant of the mapzB!B is simply identified as the originA0 of ourinitial schemeX [ Y [Z (recallB is the union of the three axes). The unionof that with the images ofX0, Y0 andZ0 is again justA0. The fiber product ofthe diagram

. zE! zB/

››

.fX0;Y0;Z0g ! fA0g/ // .� WE!B/

can be viewed asfQ1; : : : ;Q6g! fP1;P2;P3g whereQ1 andQ2 are mappedto P1 and so on (rememberE was the disjoint union of the coordinate axes ofX , of Y , and of respectivelyZ, so zE has six components and thus six origins).Thus we have the diagram

fQ1; : : : ;Q6g //

››

zE

››

fP1;P2;P3g((

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

//

››

zB&&

L

L

L

L

L

L

L

L

L

L

L

L

L

››

fX0;Y0;Z0g // E

fA0g //

((

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

B&&

�

L

L

L

L

L

L

L

L

L

L

L

L

L

which we can combine with previous diagrams to construct a cubic hyperreso-lution.


REMARK 2.17. It is possible to find a cubic hyperresolution for the three coor-dinate planes inA3 in a different way. Suppose thatS is the union of the threecoordinate planes (X , Y , andZ) of A3. Consider the 2 or ˜

C

2scheme defined

by the diagram below, where the dotted arrows are those in˜C

2but not in˜2.

X \Y \Z //

››

Y \Z

››

X \Y&&

N

N

N

N

N

N

N

N

N

N

N

//

››

Y&&

N

N

N

N

N

N

N

N

N

N

N

N

››

X \Z // Z

X //

&&

N

N

N

N

N

N

N

N

N

N

N

N

X [Y [Z&&

One can verify that this is also a cubic hyperresolution ofX [Y [Z.

Now we discuss sheaves on diagrams of schemes, as well as the related notionsof push-forward and its right derived functors.

DEFINITION 2.18 [GNPP88, I, 5.3–5.4]. LetX˝

be anI -scheme (or even anI -topological space). We define asheaf (or pre-sheaf) of abelian groupsF ˝ onX

˝to be the following data:

(i) A sheaf (pre-sheaf)F i of abelian groups overXi , for all i 2ObI , and(ii) An X�-morphism of sheavesF� W F i ! .X�/�F j for all morphisms� W

i ! j of I , required to be compatible in the obvious way.

Given a morphism of diagrams of schemesf˝W X

˝! Y

˝one can construct a

push-forward functor for sheaves onX˝.

DEFINITION 2.19 [GNPP88, I, 5.5]. LetX˝be anI -scheme,Y

˝aJ -scheme,F ˝

a sheaf onX˝, andf

˝WX

˝! Y

˝a morphism of diagrams of schemes. We define

.f˝/�F ˝ in the following way. For eachj 2ObJ we define

..f˝/�F ˝/j D lim

�.Y�/�.fi�F i/

where the inverse limit traverses all pairs.i; �/where� Wf .i/!j is a morphismin J op.

REMARK 2.20. In many applications,J will simply be the categoryf0g withone object and one morphism (for example, cubic hyperresolutions of schemes).In that case one can merely think of the limit as traversingI .

REMARK 2.21. One can also define a functorf �, show that it has a right adjointand that that adjoint isf� as defined above [GNPP88, I, 5.5].


DEFINITION 2.22 [GNPP88, I, Section 5]. LetX˝andY

˝be diagrams of topo-

logical spaces overI andJ respectively, W I ! J a functor,f˝W X

˝! Y

˝

a˚-morphism of topological spaces. IfG˝ is a sheaf overY˝

with values in acomplete categoryC, one denotes byf �

˝

G � the sheaf overX˝defined by

.f �

˝

G˝/i D f �

i .G˚.i//;

for all i 2ObI . One obtains in this way a functor

f �

˝

W Sheaves.Y˝;C/! Sheaves.X

˝;C/

Given anI -schemeX˝, one can define the category of sheaves of abelian groups

Ab.X˝/ on X

˝and show that it has enough injectives. Next, one can even define

the derived categoryDC.X˝;Ab.X

˝// by localizing bounded below complexes

of sheaves of abelian groups onX˝

by the quasi-isomorphisms (those that arequasi-isomorphisms on eachi 2 I ). One can also show that.f

˝/� as defined

above is left exact so that it has a right derived functorR.f˝/� [GNPP88, I,

5.8-5.9]. In the case of a cubic hyperresolution of a schemef WX˝! X ,

R..f˝/�F ˝/DR lim

�.Rfi�F i/

where the limit traverses the categoryI of X˝.

Final remark. We end our excursion into the world of hyperresolutions here.There are many other things to work out, but we will leave themto the interestedreader. Many “obvious” statements need to be proved, but most are relativelystraightforward once one gets comfortable using the appropriate language. Forthose and many more statements, including the full details of the constructionof the Du Bois complex and many applications, the reader is encouraged to read[GNPP88].

Acknowledgements

We thank Kevin Tucker, Zsolt Patakfalvi and the referee for reading earlierdrafts and making helpful suggestions for improving the presentation.

References

[Amb98] F. Ambro: The locus of log canonical singularities, preprint, 1998.arXiv

math.AG/9806067

[Bou87] J.-F. Boutot:Singularites rationnelles et quotients par les groupes reductifs,Invent. Math.88 (1987), no. 1, 65–68.MR 88a:14005

[BH93] W. Bruns and J. Herzog:Cohen-Macaulay rings, Cambridge Studies in Ad-vanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993.MR

95h:13020


[Car85] J. A. Carlson:Polyhedral resolutions of algebraic varieties, Trans. Amer.Math. Soc.292(1985), no. 2, 595–612.MR 87i:14008

[DB81] P. Du Bois:Complexe de de Rham filtre d’une variete singuliere, Bull. Soc.Math. France109(1981), no. 1, 41–81.MR 82j:14006

[DJ74] P. Dubois and P. Jarraud:Une propriete de commutation au changement de basedes images directes superieures du faisceau structural, C. R. Acad. Sci. Paris Ser. A279(1974), 745–747.MR0376678 (51 #12853)

[Dur79] A. H. Durfee:Fifteen characterizations of rational double points and simplecritical points, Enseign. Math. (2)25 (1979), no. 1-2, 131–163.MR 80m:14003

[Elk78] R. Elkik: Singularites rationnelles et deformations, Invent. Math.47 (1978),no. 2, 139–147.MR501926 (80c:14004)

[Elk81] R. Elkik: Rationalite des singularites canoniques, Invent. Math.64 (1981),no. 1, 1–6.MR621766 (83a:14003)

[Fed83] R. Fedder:F -purity and rational singularity, Trans. Amer. Math. Soc.278(1983), no. 2, 461–480.MR701505 (84h:13031)

[FW89] R. Fedder and K. Watanabe:A characterization ofF -regularity in terms ofF -purity, Commutative algebra (Berkeley, CA, 1987), Math. Sci. Res.Inst. Publ.,vol. 15, Springer, New York, 1989, pp. 227–245.MR 91k:13009

[Fuj08] O. Fujino:Theory of non-lc ideal sheaves: basic properties, Kyoto J. Math.50(2010), no. 2, 225–245.2666656

[GKKP10] D. Greb, S. Kebekus, S. J. Kovacs, and T. Peternell:Differential forms onlog canonical spaces, preprint, 2010.arXiv:1003.2913v3

[GT80] S. Greco and C. Traverso:On seminormal schemes, Compositio Math.40(1980), no. 3, 325–365.MR571055 (81j:14030)

[GNPP88] F. Guillen, V. Navarro Aznar, P. Pascual Gainza, and F. Puerta:Hyperre-solutions cubiques et descente cohomologique, Lecture Notes in Mathematics, vol.1335, Springer-Verlag, Berlin, 1988, Papers from the Seminar on Hodge–DeligneTheory held in Barcelona, 1982.MR 90a:14024

[Har98] N. Hara:A characterization of rational singularities in terms of injectivity ofFrobenius maps, Amer. J. Math.120(1998), no. 5, 981–996.MR 99h:13005

[Har05] N. Hara:A characteristicp analog of multiplier ideals and applications,Comm. Algebra33 (2005), no. 10, 3375–3388.MR 2006f:13006

[HW02] N. Hara and K.-I. Watanabe:F-regular and F-pure rings vs. log terminal andlog canonical singularities, J. Algebraic Geom.11(2002), no. 2, 363–392.MR 2002k:

13009

[HY03] N. Hara and K.-I. Yoshida:A generalization of tight closure and multiplierideals, Trans. Amer. Math. Soc.355(2003), no. 8, 3143–3174 (electronic).MR 2004i:

13003


[Har66] R. Hartshorne:Residues and duality, Lecture notes of a seminar on the work ofA. Grothendieck, given at Harvard 1963/64. With an appendixby P. Deligne. LectureNotes in Mathematics, No. 20, Springer-Verlag, Berlin, 1966. MR0222093 (36 #5145)

[Har77] R. Hartshorne:Algebraic geometry, Springer-Verlag, New York, 1977, Gradu-ate Texts in Mathematics, No. 52.MR0463157 (57 #3116)

[Hir64] H. Hironaka:Resolution of singularities of an algebraic variety over a field ofcharacteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2)79 (1964),205–326.MR0199184 (33 #7333)

[HH90] M. Hochster and C. Huneke:Tight closure, invariant theory, and the Briancon-Skoda theorem, J. Amer. Math. Soc.3 (1990), no. 1, 31–116.MR 91g:13010

[HH09] M. Hochster and C. Huneke:Tight closure in equal characteristic zero, Apreprint of a manuscript,http://www.math.lsa.umich.edu/hochster/dk.ps,2009.

[HR76] M. Hochster and J. L. Roberts:The purity of the Frobenius and local cohomol-ogy, Advances in Math.21 (1976), no. 2, 117–172.MR0417172 (54 #5230)

[Hun96] C. Huneke:Tight closure and its applications, CBMS Regional ConferenceSeries in Mathematics, vol. 88, Published for the Conference Board of the Mathe-matical Sciences, Washington, DC, 1996, With an appendix byMelvin Hochster.MR

96m:13001

[Ish85] S. Ishii:On isolated Gorenstein singularities, Math. Ann.270 (1985), no. 4,541–554.MR776171 (86j:32024)

[Ish86] S. Ishii:Small deformations of normal singularities, Math. Ann.275 (1986),no. 1, 139–148.MR849059 (87i:14003)

[Ish87] S. Ishii: Isolated Q-Gorenstein singularities of dimension three, Complexanalytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam,1987, pp. 165–198.MR894292 (89d:32016)

[Kaw07] M. Kawakita: Inversion of adjunction on log canonicity, Invent. Math.167(2007), no. 1, 129–133.2264806 (2008a:14025)

[KSB88] J. Kollar and N. I. Shepherd-Barron:Threefolds and deformations of surfacesingularities, Invent. Math.91 (1988), no. 2, 299–338.MR 88m:14022

[Kol95] J. Kollar: Shafarevich maps and automorphic forms, M. B. Porter Lectures,Princeton University Press, Princeton, NJ, 1995.MR 96i:14016

[Kol96] J. Kollar: Rational curves on algebraic varieties, Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveysin Mathematics,vol. 32, Springer-Verlag, Berlin, 1996.MR 98c:14001

[Kol07] J. Kollar: Two examples of surfaces with normal crossing singularities, pre-print, 2007.arXiv 0705.0926v2

[Kol08a] J. Kollar:Hulls and husks, preprint, 2008.arXiv 0805.0576v2

[Kol08b] J. Kollar:Semi log resolution, preprint, 2008.arXiv 0812.3592


[KK10] J. Kollar and S. J. Kovacs:Log canonical singularities are Du Bois, J. Amer.Math. Soc.23 (2010), no. 3, 791–813.doi:10.1090/S0894-0347-10-00663-6

[KM98] J. Kollar and S. Mori:Birational geometry of algebraic varieties, CambridgeTracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998,With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998Japanese original.MR1658959 (2000b:14018)

[Kol92] J. Kollar et. al:Flips and abundance for algebraic threefolds, Societe Mathe-matique de France, Paris, 1992, Papers from the Second Summer Seminar onAlgebraic Geometry held at the University of Utah, Salt LakeCity, Utah, August1991, Asterisque No. 211 (1992).MR 94f:14013

[Kov99] S. J. Kovacs: Rational, log canonical, Du Bois singularities: on the con-jectures of Kollar and Steenbrink, Compositio Math.118 (1999), no. 2, 123–133.MR1713307 (2001g:14022)

[Kov00a] S. J. Kovacs:A characterization of rational singularities, Duke Math. J.102(2000), no. 2, 187–191.MR1749436 (2002b:14005)

[Kov00b] S. J. Kovacs:Rational, log canonical, Du Bois singularities. II. Kodairavanishing and small deformations, Compositio Math.121 (2000), no. 3, 297–304.MR1761628 (2001m:14028)

[KSS10] S. J. Kovacs, K. Schwede, and K. E. Smith:The canonical sheaf of Du Boissingularities, Advances in Mathematics224(2010), no. 4, 1618 – 1640.

[Laz04] R. Lazarsfeld:Positivity in algebraic geometry. II, Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveysin Mathematics [Re-sults in Mathematics and Related Areas. 3rd Series. A Seriesof Modern Surveys inMathematics], vol. 49, Springer-Verlag, Berlin, 2004, Positivity for vector bundles,and multiplier ideals.MR2095472 (2005k:14001b)

[MS91] V. B. Mehta and V. Srinivas:NormalF -pure surface singularities, J. Algebra143(1991), no. 1, 130–143.MR 92j:14044

[MS97] V. B. Mehta and V. Srinivas:A characterization of rational singularities, AsianJ. Math.1 (1997), no. 2, 249–271.MR 99e:13009

[PS08] C. A. M. Peters and J. H. M. Steenbrink:Mixed Hodge structures, Ergebnisseder Mathematik und ihrer Grenzgebiete. 3. Folge. A Series ofModern Surveys inMathematics [Results in Mathematics and Related Areas. 3rdSeries. A Series ofModern Surveys in Mathematics], vol. 52, Springer-Verlag,Berlin, 2008.MR2393625

[Rei87] M. Reid:Young person’s guide to canonical singularities, Algebraic geometry,Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. PureMath., vol. 46, Amer.Math. Soc., Providence, RI, 1987, pp. 345–414.MR 89b:14016

[Rei97] M. Reid:Chapters on algebraic surfaces, Complex algebraic geometry (ParkCity, UT, 1993), IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence, RI,1997, pp. 3–159.MR1442522 (98d:14049)

[Sai00] M. Saito:Mixed Hodge complexes on algebraic varieties, Math. Ann. 316(2000), no. 2, 283–331.MR1741272 (2002h:14012)


[Sai09] M. Saito:On the Hodge filtration of Hodge modules, Mosc. Math. J.9 (2009),no. 1, 161–191.

[Sch82] P. Schenzel:Applications of dualizing complexes to Buchsbaum rings, Adv. inMath.44 (1982), no. 1, 61–77.MR 83j:13011

[Sch07] K. Schwede:A simple characterization of Du Bois singularities, Compos.Math.143(2007), no. 4, 813–828.MR2339829

[Sch08] K. Schwede:Centers ofF -purity, Math. Z. 265 (2010), no. 3, 687–714.MR2644316

[Sch09] K. Schwede:F -injective singularities are Du Bois, Amer. J. Math.131(2009),no. 2, 445–473.MR2503989

[ST08] K. Schwede and S. Takagi:Rational singularities associated to pairs, MichiganMath. J.57 (2008), 625–658.

[Smi97] K. E. Smith:F -rational rings have rational singularities, Amer. J. Math.119(1997), no. 1, 159–180.MR 97k:13004

[Smi00] K. E. Smith:The multiplier ideal is a universal test ideal, Comm. Algebra28(2000), no. 12, 5915–5929, Special issue in honor of Robin Hartshorne.MR 2002d:

13008

[Smi01] K. E. Smith:Tight closure and vanishing theorems, School on VanishingTheorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect.Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 149–213.MR1919458 (2003f:13005)

[Ste83] J. H. M. Steenbrink:Mixed Hodge structures associated with isolated singular-ities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40,Amer. Math. Soc., Providence, RI, 1983, pp. 513–536.MR 85d:32044

[Ste85] J. H. M. Steenbrink:Vanishing theorems on singular spaces, Asterisque (1985),no. 130, 330–341, Differential systems and singularities (Luminy, 1983).MR 87j:

14026

[Swa80] R. G. Swan:On seminormality, J. Algebra67 (1980), no. 1, 210–229.MR 82d:

13006

[Tak04] S. Takagi:An interpretation of multiplier ideals via tight closure, J. AlgebraicGeom.13 (2004), no. 2, 393–415.MR 2005c:13002

[Tak08] S. Takagi:A characteristicp analogue of plt singularities and adjoint ideals,Math. Z.259(2008), no. 2, 321–341.MR 2009b:13004

[TW04] S. Takagi and K.-i. Watanabe:On F-pure thresholds, J. Algebra282 (2004),no. 1, 278–297.MR2097584 (2006a:13010)

[Wat87] K.-i. Watanabe:On plurigenera of normal isolated singularities. II, Complexanalytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam,1987, pp. 671–685.MR 89h:32029


SANDOR J. KOVACS

UNIVERSITY OF WASHINGTON

DEPARTMENT OFMATHEMATICS

SEATTLE, WA 98195UNITED STATES

[email protected]

KARL E. SCHWEDE


THE PENNSYLVANIA STATE UNIVERSITY

UNIVERSITY PARK , PA 16802UNITED STATES

[email protected]


Elliptic genera, real algebraic varietiesand quasi-Jacobi forms

ANATOLY LIBGOBER

ABSTRACT. We survey the push-forward formula for elliptic class and variousapplications obtained in the papers by L. Borisov and the author. We then dis-cuss the ring of quasi-Jacobi forms which allows to characterize the functionswhich are the elliptic genera of almost complex manifolds and extension ofOchanine elliptic genus to certain singular real algebraicvarieties.

Introduction

Interest in the elliptic genus of complex manifolds stems from its appearancein a wide variety of geometric and topological problems. Theelliptic genus isan invariant of the complex cobordism class modulo torsion,and hence dependsonly on the Chern numbers of the manifold. On the other hand, the elliptic genusis a holomorphic function defined onC�H, whereH is the upper half plane.In one heuristic approach, the elliptic genus is an index of an operator on theloop space (see [53]) and as such it has counterparts defined for C 1, orientedor Spin manifolds: these were in fact studied before the complex case [43].

The elliptic genus comes up in the study of the geometry and topology of loopspaces and, more specifically, of the chiral de Rham complex [41]; in the study ofinvariants of singular algebraic varieties [8] — in particular orbifolds; and, morerecently, in the study of Gopakumar–Vafa and Nekrasov conjectures [38; 25].It is closely related to the fast developing subject of elliptic cohomology [46].There are various versions of the elliptic genus, includingthe equivariant, higherelliptic genus obtained by twisting by cohomology classes of the fundamentalgroup, the elliptic genus of pairs and the orbifold ellipticgenus. There is inter-

The author was partially supported by an NSF grant.

95

96 ANATOLY LIBGOBER

esting connection with singularities of weighted homogeneous polynomials —the so-called Landau–Ginzburg models.

We shall review several recent developments on the ellipticgenus; we referthe reader [7] for details on earlier results. Then we shall focus on certain aspectsof the elliptic genus: its extension to real singular varieties and its modularityproperty (or the lack of it). Extensions of the elliptic genus to real singularvarieties were suggested by B. Totaro [48]; our approach is based on the push-forward formula for the elliptic class used in [8] to extend elliptic genus fromsmooth to certain singular complex projective varieties.

In Section 1A we discuss this push-forward formula, which appears as themain technical tool in many applications mentioned later. The rest of Section 1discusses the relation with other invariants and series of applications based onthe material in [9; 8; 6]. It includes a discussion of a relation between ellip-tic genus andE-function, applications to the McKay correspondence, ellipticgenera of non-simply connected manifolds (higher ellipticgenera) and general-izations of a formula of R. Dijkgraaf, D. Moore, E. Verlinde,and H. Verlinde.(Other applications in the equivariant context are discussed in R. Waelder’s paperin this volume.) The proof of independence of resolutions ofthe elliptic genus(according to our definition) for certain real algebraic varieties is given later, inSection 3B.

Section 2 deals with modularity properties of the elliptic genus. In the Calabi–Yau cases (of pairs, orbifolds, etc.) the elliptic genus is aweak Jacobi form; seedefinition below. Also it is important to have a description of the elliptic genusin non-Calabi–Yau situations not just as a function onC�H but as an elementof a finite-dimensional algebra of functions. It turns out that in the absence of aCalabi–Yau condition the elliptic genus belong to a very interesting algebra offunctions onC�H, which we call the algebra of quasi-Jacobi forms, and whichis only slightly bigger than the algebra of weak Jacobi forms. This algebra ofquasi-Jacobi forms is a counterpart of quasimodular forms [31] and is relatedto the elliptic genus in the same way as quasimodular forms are related to theWitten genus [55]. The algebra of quasi-Jacobi forms is generated by certaintwo-variable Eisenstein series, masterfully reviewed by A. Weil in [51], andhas many properties parallel to the properties in quasimodular case. A detaileddescription of the properties of quasi-Jacobi forms appears to be absent in theliterature, so we discuss the algebra of such forms in Section 2 — see its in-troduction. We conclude Section 2 with a discussion of differential operatorsRankin–Cohen brackets on the space of Jacobi forms.

Finally in Section 3 we construct an extension of the Ochanine genus to realalgebraic varieties with certain class of singularities. This extends results ofTotaro [48].

ELLIPTIC GENERA, REAL ALGEBRAIC VARIETIES AND QUASI-JACOBI FORMS 97

For the readers’ convenience we give ample references to prior work on ellip-tic genus, where more detailed information can be obtained.Section 2, dealingwith quasi-Jacobi forms, can be read independently of the rest of the paper.

1. Elliptic genus

1A. Elliptic genus of singular varieties and push-forward formulas. Let X

be a projective manifold. We shall use the Chow groupsA�.X / with complexcoefficients (see [23]). LetF the ring of functions onC �H whereH is theupper half-plane. The elliptic class ofX is an element inA�.X /˝C F given by

ELL.X /DY

i

xi

��

xi

2� i � z; ��

��

xi

2� i ; �� ŒX �; (1-1)

where

�.z; �/D q18 .2 sin�z/

lD1Y

lD1

.1� ql /

lD1Y

lD1

.1� qle2� iz/.1� qle�2� iz/ (1-2)

is the Jacobi theta function considered as an element inF with q D e2� i� [12],thexi are the Chern roots of the tangent bundle ofX , andŒX � is the fundamentalclass ofX . The componentEll.X / in A0.X /D F is the elliptic genus ofX .

The components of (1-1) in each degree, evaluated on a class in A�.X /, arelinear combinations of symmetric functions inci : that is, the Chern classes ofX .In particular,Ell.X / depends only on the class ofX in the ring˝U ˝Q ofunitary cobordisms.

The homomorphism U ˝ Q ! F taking X to Ell.X / can be describedwithout reference to theta functions. LetM3;A1

be the class of complex analyticspaces “having onlyA1-singularities in codimension three”, that is, having onlysingularities of the following type: the singular set SingX of X 2M3;A1

is amanifold such that dimC SingX D dimX � 3 and for an embeddingX ! Y

whereY is a manifold and a transversalH to SingX in Y , the pair

.H \X;H \SingX /

is analytically equivalent to the pair.C4;H0/, whereH0 is given byx2Cy2C

z2 C w2 D 0. EachX 2 M3;A1admits two small resolutionsQX1 ! X and

QX2! X in which the exceptional set is a fibration over SingX with the fiberP1. One says that the manifolds underlying the resolutions areobtained fromeach other by a classical flop.

THEOREM 1.1. (cf . [47]) The kernel of the homomorphismEll W˝U ˝Q!F

taking an almost complex manifoldX to its elliptic genusEll.X / is the idealgenerated by the classes of differencesQX1 � QX2 of two small resolutions of avariety inM3;A1

.

98 ANATOLY LIBGOBER

More generally one can fix a class of singular spaces and a typeof resolutionsand consider the quotient of U ˝ Q by the ideal generated by differencesof manifolds underlying resolutions of the same analytic space. The quotientmap by this ideal U ˝ Q ! R provides a genus and hence a collection ofChern numbers (linear combination of Chern monomialsci1

: : : � cikŒX �, with

P

is D dimX ), which can be made explicit via Hirzebruch’s procedure with agenerating series [28]. These are the Chern numbers which can be defined for thechosen class of singular varieties and chosen class of resolutions. The ideal inTheorem 1.1, it turns out, corresponds to a much larger classes of singular spacesand resolutions. This method of defining Chern classes of singular varieties is anextension of the philosophy underlying a question of Goresky and McPherson[26]: Which Chern numbers can be defined via resolutions independently of theresolution?

DEFINITION 1.2. An analytic spaceX is calledQ-Gorenstein if the divisorD of a meromorphic formdf1 ^ � � � ^ dfdimX is such that for somen 2 Z thedivisor nD in locally principal (i.e.,KX is Q-Cartier). In particular, for anycodimension-one componentE of the exceptional divisor of a map� W QX !X ,the multiplicity aE DmultE��.KX / is well defined and a singularity is calledlog-terminal if there is a resolution� such thatK QX

D ��.KX /CP

aEE andaE > �1. A resolution is called crepant ifaE D 0.

THEOREM1.3. ([8])The kernel of the elliptic genusU ˝Q!F is generatedby the differences ofQX1� QX2 of manifolds underlying crepant resolutions of thesingular spaces withQ-Gorenstein singularities admitting crepant resolutions.

The proof of Theorem 1.3 is based on an extension of the elliptic genusEll.X /

of manifolds to the elliptic genus of pairsEll.X;D/, whereD is a divisor onXhaving normal crossings as the only singularities. This is similar to the situationin the study of motivicE-functions of quasiprojective varieties [2; 40]. In fact,other problems such as the the study of McKay correspondence[9] suggest amotivation for looking at triples.X;D;G/, whereX is a normal variety,G isa finite group acting onX and to introduce the elliptic classELL.X;D;G/

(see again [9]). More precisely, letD DP

aiDi be aQ-divisor, with theDi

irreducible andai 2 Q. The pair.X;D/ is called Kawamata log-terminal (klt)[35] if KX CD is Q-Cartier and there is a birational morphismf W Y ! X ,whereY is smooth and is the union of the proper preimages of componentsof D, and the components of the exceptional setE D

S

J Ej form a normalcrossing divisor such thatKY Df

�.KXCP

aiDi/CP

j Ej , where j >�1.(HereKX ;KY are the canonical classes ofX andY .) The triple.X;D;G/,whereX is a nonsingular variety,D is a divisor andG is a finite group ofbiholomorphic automorphisms is calledG-normal [2; 9] if the components of


D form a normal crossings divisor and the isotropy group of anypoint actstrivially on the components ofD containing this point.

DEFINITION 1.4 [9, Definition 3.2]. Let.X;E/ be a Kawamata log terminalG-normal pair (in particular,X is smooth andD is a normal crossing divisor)and letED�

P

k2K ıkEk . Theorbifold elliptic classof .X;E;G/ is the classin A�.X;Q/ given by

ELLorb.X;E;GI z; �/ WD

1

jGj

X

g;hghDhg

X

X g;h

ŒX g;h�Y

�.g/D0�.h/D0

x�Y

�

��

x�

2� i C�.g/� ��.h/� z�

��

x�

2� i C�.g/� ��.h/� e2� i�.h/z

�Y

k

��

ek

2� i C "k.g/� "k.h/� � .ıkC1/z�

��

ek

2� i C "k.g/� "k.h/� � z�

�.�z/

�.�.ık C 1/z/e2� iık"k.h/z : (1-3)

whereX g;h denotes an irreducible component of the fixed set of the commutingelementsg and h and ŒX g;h� denotes the image of the fundamental class inA�.X /. The restriction ofTX to X g;h splits into linearized bundles accordingto the (Œ0; 1/-valued) characters� of hg; hi, which are sometimes denoted by�W , whereW is a component of the fixed-point set. Moreover,ek D c1.Ek/

and"k is the character ofO.Ek/ restricted toX g;h if Ek containsX g;h, and iszero otherwise.

One would like to define the elliptic genus of a Kawamata log-terminal pair.X0;D0/ as (1-3) calculated for aG-equivariant resolution.X;E/! .X0;D0/.Independence of (1-3) of resolution and the proof of (1.3) both depend on thefollowing push-forward formula:

THEOREM 1.5. Let .X;E/ be a Kawamata log-terminalG-normal pair and letZ be a smoothG-equivariant locus inX which is normal crossing toSuppE.Letf W OX ! X denote the blowup ofX alongZ. Define

OE D�X

k

ık OEk � ı Excf;

where OEk is the proper transform ofEk andı is determined fromK OXC OE D

f �.KX CE/. Then. OX ; OE/ is a Kawamata log-terminalG-normal pair and

f�ELLorb. OX ; OE;GI z; �/D ELLorb.X;E;GI z; �/: (1-4)

Independence from the resolution is a consequence of the weak factorizationtheorem [1] and of Theorem 1.5; Theorem 1.3 follows since both ELL.X1/

and ELL.X2/ coincide with the elliptic genus of the pair. QX ; QD/, where QXis a resolution ofX dominating bothX1 and X2 (hereD D K QX =X

; see [8,

100 ANATOLY LIBGOBER

Proposition 3.5] and also [52]. For a discussion of the orbifold elliptic genus onorbifolds more general than just global quotients see [20].

1B. Relation to other invariants. V. Batyrev defined in [2], for aG-normaltriple .X;D;G/, anE-function Eorb.X;D;G/ depending on the Hodge theo-retical invariants. (There is also a motivic version; see [2; 40].) Firstly for aquasiprojective algebraic varietyW one sets, as in [2, Definition 2.10],

E.W;u; v/D .�1/iX

p;q

dimGrpF

GrpCqW

.H ic .W;C//upvq; (1-5)

whereF andW are the Hodge and weight filtrations of Deligne’s mixed Hodgestructure [16; 17]. In particular,E.W; 1; 1/ is the topological Euler characteris-tic of W (with compact support). IfW is compact one then obtains Hirzebruch’s�y-genus [28]:

�y.W /DX

i;j

.�1/q dimH q.˝pW/yp; (1-6)

for v D �1, u D y, and hence the arithmetic genus, signature and so on arespecial values of (1-5). Secondly, for aG-normal pair as in Definition 1.4 onestratifiesD D

S

k2K Dk by strataDı

JDT

j2J Dj �S

k2K�J Dk , for J �K

(the intersection being set toX if J D?), and defines

E.X;D;G;u; v/

DX

fgg

W �X g

.uv/P

"Di.g/.ıi C1/

X

J �Kg

Y

j2J

uv�1

.uv/ıj C1�1E.W \Dı

J =C.g;J //; (1-7)

whereC.g;J / is the subgroup of the centralizer ofg leavingT

j2J Dj invariant.One shows that for a Kawamata log-terminal pair.X0;D0/ the E-function

E.X;D;G/ of a resolution does not depend on the latter but only onX0, D0

andG. Hence (1-7) yields an invariant of Kawamata log-terminalG-pairs. Therelation withEll.X;D;G/ is the following (Proposition 3.14 of [9]):

lim�!i1

Ell.X;D;G; z; �/D y�

12

dimXE.X;D;G;y; 1/; (1-8)

wherey D exp.2� iz/. In particular, in the nonequivariant smooth case theelliptic genus forq! 0 specializes into the Hirzebruch�y genus (1-6).

On the other hand, in the nonsingular case, Hirzebruch [29; 30] and Witten[53] defined elliptic genera of complex manifolds which are given by modularforms for the subgroup�0.n/ on leveln in SL2.Z/, provided the canonical classof the manifold in question is divisible byn.

These genera are of course combinations of Chern numbers, but for nD2 oneobtains a combination of Pontryagin classes; i.e., an invariant that depends onlyon the underlying smooth structure, rather than the (almost) complex structure.


This genus was first introduced by S. Ochanine; see [43] and Section 3. Theselevel-n elliptic genera coincide, up to a dimensional factor, with the specializa-tion z D .˛� C b/=n, for appropriate ; ˇ 2 Z specifying particular Hirzebruchlevel n elliptic genus; see Proposition 3.4 of [6].

1C. Application: The McKay correspondence for the elliptic genus. Theclassical McKay correspondence is a relation between the representations ofthe binary dihedral groupsG � SU.2/ (which are classified according to theroot systems of typeAn;Dn;E6;E7;E8) and the irreducible components of theexceptional set of the minimal resolution ofC2=G. In particular, the number ofconjugacy classes inG is the same as the number of irreducible components ofthe minimal resolution. The latter is a special case of the relation between theEuler characteristice. eX=G/ of a crepant resolution of the quotientX=G of acomplex manifoldX by an action of a finite groupG and the data of the actionon X :

e. eX=G/DX

g;hghDhg

e.X g;h/: (1-9)

A refinement of this relation for Hodge numbers and motives isgiven in [2;19; 40]. WhenX is projective one has a refinement in which the Euler charac-teristic of the manifold in (1-9) is replaced by the ellipticgenus of Kawamatalog-terminal pairs. More generally, one has the following push-forward formula:

THEOREM 1.6. Let .X IDX / be a Kawamata log-terminal pair which is invari-ant under an effective action of a finite groupG on X . Let WX ! X=G bethe quotient morphism. Let .X=GIDX=G/ be the quotient pair in the sense thatDX=G is the unique divisor onX=G such that �.KX=GCDX=G/DKX CDX

(see Definition 2.7 in [9]).Then

�ELLorb.X;DX ;GI z; �/D ELL.X=G;DX=G I z; �/:

In particular, for the components of degree zero one obtains

El lorb.X;DX ;G; z; �/DEll.X=G;DX=G; z; �/: (1-10)

WhenX is nonsingular andX=G admits a crepant resolutioneX=G!X=G, forq D 0 one obtains�y. eX=G/D �y

orb.X;G/ and hence fory D 1 one recovers(1-9).

1D. Application: Higher elliptic genera and K-equivalences. Another appli-cation of the push-forward formula in Theorem 1.5 is the invariance of higherelliptic genera under K-equivalences. A question posed in [44], and answeredin [3], concerns the higher arithmetic genus�˛.X / of a complex manifoldX


corresponding to a cohomology class˛ 2H �.�1.X /;Q/ and defined asZ

X

T dX [f�.˛/; (1-11)

wheref W X ! B.�1.X // is the classifying map fromX to the classifyingspace of the fundamental group ofX . It asks whether the higher arithmeticgenus�˛.X / is a birational invariant. This question is motivated by Novikov’sconjecture: the higher signatures (i.e., the invariant defined for topological man-ifold X by (1-11) with the Todd class replaced by theL-class) are homotopyinvariant [15]. The higher�y-genus defined by (1-11) with the Todd class re-placed by Hirzebruch’s�y class [28] comes into the correction terms describingthe nonmultiplicativity of�y in topologically locally trivial fibrations� WE!B

of projective manifolds with nontrivial action of�1.B/ on the cohomology ofthe fibers of�. See [11] for details.

Recall that two manifoldsX1;X2 are calledK-equivalent if there is a smoothmanifold QX and a diagram

zX

X1

�1

�X2

�2- (1-12)

in which �1 and�2 are birational morphisms and��

1.KX1

/ and��

2.KX2

/ arelinearly equivalent.

The push-forward formula (1.5) leads to:

THEOREM 1.7. For any˛ 2H �.B�;Q/ the higher elliptic genus

.ELL.X /[f �.˛/; ŒX �/

is an invariant ofK-equivalence. Moreover, if .X;D;G/ and. OX ; OD;G/ areG-normal and Kawamata log-terminal and if� W. OX ; OD/!.X;D/ isG-equivariantsuch that

��.KX CD/DK OXC OD; (1-13)

thenEll˛. OX ; OD;G/DEll˛.X;D;G/:

In particular the higher elliptic genera(and hence the higher signatures andOA-genus) are invariant for crepant morphisms. The specialization into the Todd

class is birationally invariant(i.e., the invariance condition(1-12)is not neededin the Todd case).

Another consequence is the possibility of defining higher elliptic genera forsingular varieties with Kawamata log-terminal singularities and forG-normalpairs.X;D/; see [10].


1E. The DMVV formula. The elliptic genus comes into a beautiful productformula for the generating series for the orbifold ellipticgenus associated withthe action of the symmetric groupSn on productsX �� X , for which the firstcase appears in [18], together with a string-theoretical explanation. A generalproduct formula for orbifold elliptic genus of triples is given in [9].

THEOREM 1.8. Let .X;D/ be a Kawamata log-terminal pair. For everyn � 0

consider the quotient of.X;D/n by the symmetric groupSn, which we willdenote by.X n=Sn;D

.n/=Sn/. Here we denote byD.n/ the sum of pullbacks ofD undern canonical projections toX . Then we have

X

n�0

pnEll.X n=Sn;D.n/=SnI z; �/D

1Y

iD1

Y

l;m

1

.1�piylqm/c.mi;l/; (1-14)

where the elliptic genus of.X;D/ isX

m�0

X

l

c.m; l/ylqm

andy D e2� iz , q D e2� i� .

It is amazing that such a simple-minded construction as the left-hand side of(1-14) leads to the Borcherds lift [4] of Jacobi forms.

1F. Other applications of the elliptic genus.In this section we point out otherinstances in which the elliptic genus plays a significant role.

The chiral de Rham complex. In [41], the authors construct for a complexmanifold X a (bi)-graded sheaf ch

Xof vertex operator algebras (with degrees

called fermionic charge and conformal weight) with the differentialdchDR

havingfermionic degree 1 and quasiisomorphic to the de Rham complex of X . Analternative construction using the formal loop space was given in [32]. Eachcomponent of fixed conformal weight has a filtration so that graded componentsare

O

n�1

.��yqn�1T �

X ˝��y�1qnTX ˝SqnT �

X ˝SqnTX / (1-15)

In particular, it follows that

Ell.X; q;y/D y�

12

dimX�.˝ch

X /D y�

12

dimX SupertraceH �.˝chX/y

J Œ0�qLŒ0�;

(1-16)whereJ Œm�;LŒn� are the operators which are part of the vertex algebra structure.The chiral complex for orbifolds was constructed in [22] andthe extension of(1-16) to orbifolds (with discrete torsion) is discussed in[39].


Mirror symmetry. The physics definition of mirror symmetry in terms of con-formal field theory suggests that for the elliptic genus, defined as an invariant ofa conformal field theory (by an expression similar to the lastterm in (1-16) —see [54]) one should have forX and its mirror partnerOX the relation

Ell.X /D .�1/dimX Ell. OX /: (1-17)

This is indeed the case [6, Remark 6.9] for mirror symmetric hypersurfaces intoric varieties in the sense of Batyrev.

Elliptic genus of Landau–Ginzburg models.The physics literature (see [34],for example) also associates to a weighted homogeneous polynomial a con-formal field theory (the Landau–Ginzburg model) and in particular the ellipticgenus. Moreover it is expected that the orbifoldized Landau–Ginzburg modelwill coincide with the conformal field theory of the hypersurface correspondingto this weighted homogeneous polynomial. In particular, one expects a certainidentity expressing equality of the orbifoldized ellipticgenus corresponding tothe weighted homogeneous polynomial (or a more general Landau Ginzburgmodel) and the elliptic genus of the corresponding hypersurface. In [42] theauthors construct a vertex operator algebra related by a correspondence of thistype to the cohomology of the chiral de Rham complex of the hypersurface inPn,and obtain in particular the expression for the elliptic genus of a hypersurface asan orbifoldization. In [27] the authors obtain an expression for the one-variableHirzebruch’s genus as an orbifoldization.

Concluding remarks. There are several other interesting issues which shouldbe mentioned in a discussion of the elliptic genus. It plays an important role inwork of J. Li, K. F. Liu and J. Zhou [38] in connection with the Gopakumar–Vafaconjecture (see also [25]). The elliptic genus was defined for proper schemeswith 1-perfect obstruction theory [21]. In fact one has welldefined cobordismclasses in U associated to such objects [14]. In the case of surfaces withnormal singularities, one can extend the definition of elliptic genus beyond log-terminal singularities [50]. The elliptic genus is centralin the study of ellipticcohomology [46]. Much of the discussion above can be extended to the equi-variant context [49]; a survey of this is given in Waelder’s paper in this volume.

2. Quasi-Jacobi forms

The Eisenstein series

ek.�/DX

.m;n/2Z2

.m;n/¤.0;0/

1

.m� C n/k; � 2H;


fails to be modular fork D 2, but the algebra generated by the functionsek.�/,k�2, called the algebra of quasimodular forms on SL2.Z/, has many interestingproperties [57]. For example, there is a correspondence between quasimodularforms and real analytic functions onH which have the same SL2.Z/ transfor-mation properties as modular forms. Moreover, the algebra of quasimodularforms has a structure ofD-module and supports an extension of Rankin–Cohenoperations on modular forms.

In this section we show that there is an algebra of functions on C�H closelyrelated to the algebra of Jacobi forms of index zero with similar properties.This algebra is generated by the Eisenstein series

P

.zC!/�n, the sum beingover elements! of a latticeW � C. It has a description in terms of real ana-lytic functions satisfying a functional equation of Jacobiforms and having otherproperties of quasimodular forms mentioned in the last paragraph. It turns outthat the space of functions onC �H generated by elliptic genera of arbitrary(possibly not Calabi–Yau) complex manifolds belong to thisalgebra of quasi-Jacobi forms.

DEFINITION 2.1. Aweak(resp. meromorphic) Jacobi form of indext 2 12

Z andweight k for a finite index subgroup of the Jacobi group� J

1D SL2.Z/ / Z2

is a holomorphic (resp. meromorphic) function� on H �C having expansionP

cn;r qn�r in q D exp.2�p�1�/ with Im � sufficiently large and satisfying

the functional equations

�

�

a� C b

c� C d;

z

c� C d

�

D .c� C d/ke2� itcz2=.c�Cd/�.�; z/;

�.�; zC�� C�/D .�1/2t.�C�/e�2� it.�2�C2�z/�.�; z/

for all elements��a

cbd

�

; 0�

and��1

001

�

; .a; b/�

in � . The algebra of Jacobi formsis the bigraded algebraJ D

L

Jt;k . and the algebra of Jacobi forms of indexzero is the subalgebraJ0 D

L

k J0;k � J .

For appropriatel a Jacobi form can be expanded in (Fourier) series inq1= l , withl depending on� . We shall need below the real analytic functions

�.z; �/Dz� Nz

� � N�and �.�/D

1

� � N�: (2-1)

They have the transformation properties

�

�

z

c� C d;

a� C b

c� C d

�

D .c� C d/�.z; �/� 2icz; (2-2)

�.zCm� C n; �/D �.z; �/Cm;

�.a� C b

c� C d/D .c� C d/2�.�/� 2ic.c� C d/: (2-3)


DEFINITION 2.2. AnAlmost meromorphic Jacobi formof weightk, index zeroand depth.s; t/ is a (real) meromorphic function inCfq1= l ; zgŒz�1; �; ��, with�;� given by (2-1), which

(a) satisfies the functional equations (2.1) of Jacobi formsof weightk and indexzero, and

(b) which has degree at mosts in � and at mostt in �.

DEFINITION 2.3. A quasi-Jacobi formis a constant term of an almost mero-morphic Jacobi form of index zero considered as a polynomialin the functions�;�; in other words, a meromorphic functionf0 on H � C such that thereexist meromorphic functionsfi;j such that eachf0C

P

fi;j�i�j is an almost

meromorphic Jacobi form.

From the algebraic independence of�;� over the field of meromorphic func-tions inq; z one deduces:

PROPOSITION2.4. F is a quasi-Jacobi of depth.s; t/ if and only if

.c� C d/�kf�

a�Cb

c�Cd;

z

c�Cd

�

DX

i�sj�t

Si;j .f /.�; z/�

cz

c�Cd

�i� c

c�Cd

�j;

f .�; zC a� C b/DX

i�s

Ti.f /.�; z/ai :

We turn to some basic examples of quasi-Jacobi forms.

DEFINITION 2.5 [51]. Consider the sequence of functions onH�C given by

En.z; �/DX

.a;b/2Z2

1

.zC a� C b/n

(These series were used in [24] under the name twisted Eisenstein series.)

The seriesEn.z; �/ converges absolutely forn� 3 and fornD 1; 2 defined via“Eisenstein summation” as

X

e. � /D lim

A!1

aDAX

aD�A

limB!1

bDBX

bD�B

. � /;

though we shall omit the subscripte. The seriesE2.z; �/ is related to the Weier-strass function as follows:

}.z; �/D1

z2C

X

.a;b/2Z

.a;b/¤0

1

.zC a� C b/2�

1

.a� C b/2

DE2.z; �/� limz!0

�

E2.z; �/�1

z2

�

:


Moreover,

en D limz!0

�

En.z; �/�1

zn

�

DX

.a;b/2Z

.a;b/¤0

1

.a� C b/n

is the Eisenstein series, in the notation of [51]. The algebra of functions ofHgenerated by the Eisenstein seriesen.�/ for n� 2 is the algebra of quasimodularforms for SL2.Z/ [55; 57].

Now we describe the algebra of quasi-Jacobi forms for the Jacobi group� J1

.

PROPOSITION2.6. The functionsEn are weak meromorphic Jacobi forms ofindex zero and weightn for n � 3. E1 is a quasi-Jacobi form of index0 weight1 and depth.1; 0/. E2 � e2 is a weak Jacobi form of index zero and weight2

andE2 is a quasi-Jacobi form of weight2, index zero and depth.0; 1/.

PROOF. The first part follows from the absolute convergence of the series (2.5)for n� 3. We have the transformation formulas

E1

�

a� C bc� C d ;z

c�Cd

�

D .c� C d/E1.�; z/C� ic

2z; (2-4)

E1.�; zCm� C n/ DE1.�; z/� 2� im; (2-5)

E2

�

a�Cb

c�Cd;

z

c�Cd

�

D .c� C d/2E2.�; z/�12� ic.c� C d/; (2-6)

E2.�; zC a� C b/ DE2.�; z/: (2-7)

Equalities (2-4) and (2-6) follow fromE1.�; z/D 1=z�P

e2k.�/z2k�1 and

E2.�; z/D1=z2CP

k.2k�1/e2kz2k�2, respectively; see [51, Chapter 3, (10)].Equality (2-7) is immediate form the definition of Eisenstein summation, while(2-5) follows from [51]. ˜

REMARK 2.7. The Eisenstein seriesek.�/, for k � 4, belong to the algebra ofquasi-Jacobi forms. Indeed, from [51, Chapter IV, (7), (35)] one has

E4 D .E2� e2/2� 5e4I E2

3 D .E2� e2/2� 15e4.E2� e2/� 35e4:

PROPOSITION2.8. The algebra of Jacobi forms(for � J1

) of index zero andweightt � 2 is generated byE2� e2;E3;E4.

A short way to show this is to notice that the ring of such Jacobi forms is iso-morphic to the ring of cobordisms of SU-manifolds modulo flops (Section 1A)via an isomorphism sending a complex manifoldX of dimensiond to Ell.X / �

.� 0.0/=�.z//d . This ring of cobordisms in turn is isomorphic toCŒx1;x2;x3�,wherex1 is the cobordism class of a K3 surface andx2;x3 are the cobordismclasses of certain four- and six-manifolds [48]. The gradedalgebraCŒE2�e2;

E3;E4� is isomorphic to the same ring of polynomials (Examples 2.14) and theclaim follows.


PROPOSITION2.9. The algebra of quasi-Jacobi forms is the algebra of func-tions onH�C generated by the functionsEn.z; �/ ande2.�/.

PROOF. The coefficient of�s for an almost meromorphic Jacobi formF.�; z/DP

i�s fi�i of depth.s; 0/ is a holomorphic Jacobi form of index zero and weight

k � s; thus, by the previous proposition, it is a polynomial inE2� e2, E3, . . . .Moreoverf0 �Es

1fs is a quasi-Jacobi form of index zero and weight at most

s�1. Hence, by induction, the ring of quasi-Jacobi forms of index zero and depth.�; 0/ can be identified withCŒE1;E2�e2;E3; : : :�. Similarly, the coefficient�t of an almost meromorphic Jacobi formF D

P

j�t

�P

fi;j�i�

�j is an almostmeromorphic Jacobi form of depth.s; 0/, andF �

�P

if�

i; s�i/Et2

has depth.s0; t 0/ with t 0 < t . The claim follows. ˜

Here is an alternative description of the algebra of quasi-Jacobi forms:

PROPOSITION2.10. The algebra of functions generated by the coefficients ofthe Taylor expansion inx of the function:

�.xC z/� 0.0/

�.x/�.z/��

1

xC

1

z

�

DX

i�1

Fixi

is the algebra of quasi-Jacobi forms forSL2.Z/.

PROOF. From [45] we have the transformation formulas

��

a�Cb

c�Cd;

z

c�Cd

�

D �.c� C d/1=2e� icz2=.c�Cd/�.�; z/;

� 0

�

a�Cb

c�Cd; 0�

D �.c� C d/3=2� 0.�; 0/;

�.�; zCm� C n/D .�1/mCne�2� imz�� im2��.�; z/:

They imply that the function

˚.x; z; �/Dx�.xC z/� 0.0/

�.x/�.z/(2-8)

satisfies the functional equations

˚.a� C b

c� C d;

x

c� C d;

z

c� C d/D e2� iczx=.c�Cd/˚.x; z; �/;

˚.x; zCm� C n; �/D e2� imx˚.x; z; �/:

(2-9)

In particular, in the expansion

d2 log˚

dx2DX

Hixi ; (2-10)


the left-hand side is invariant under the transformations in (2-9) and the coeffi-cient Hi is a Jacobi form of weighti and index zero for anyi . Moreover thecoefficientsFi in ˚.x; z; �/D 1C

P

Fi.z; �/xi are polynomials inF1 and the

Hi . What remains to show is that theEi determineF1 and theHi , for i � 1,and vice versa.

Recall thatEi has index zero (is invariant with respect to shifts) and weight i .We shall use the expressions

˚.x; z; �/DxC z

zexp

�

X

k>0

2

k!

�

xk C zk � .xC z/k�

Gk.�/

�

; (2-11)

where

Gk.�/D�Bk

2kC

1X

lD1

X

d jl

.dk�1/ql I (2-12)

see [56]. On the other hand, from [51, III.7 (10)] we have

En.z; �/D1

znC .�1/n

1X

2m�n

�

2m�1

n�1

��

e2mz2m�n; (2-13)

where

e2m DX0

�

1

m�Cn

�2mD

2.2�p�1/k

.k � 1/!Gk for k D 2mI (2-14)

see [51, III.7] and [55, p. 220]. We have

d2 log˚.x; z; �/

dx2

DX

i�1

.�1/i ixi�1

ziC1CX

i�2

2

.i�2/!

�

xi�2� .xC z/i�2�

Gi.�/: (2-15)

Now, using (2-14) and identities with binomial coefficients, we obtain for thecoefficient ofxl�2 for l � 2 in the Laurent expansion the value

.�1/l�1.l�1/

zl�

X

i�2;i>l

2

.i � 2/!

�

i�2

l�2

�

zi�lGi.�/

D.�1/l�1.l�1/

zl�1�X

i�2i>l

1

.2�p�1/i

.i � 1/�

i�2

l�2

�

zi�lei

D.�1/l�1.l�1/

zl� .l � 1/

1

.2�p�1/l

X

i�2i>l

�

i�1

l�1

�

ei

�

z

2�p�1

�i�l

(2-16)


This yields

Hl�2.2�p�1z; �/D .�1/l�1 .l � 1/

.2�p�1/l

.El � el /;

and the claim follows since formula (15) in [56] yields

F1.z; �/D1

z� 2

X

r�0

GrC1

zr

r !D

1

z�

1

.2�p�1/

X

r�0

er

�

z

2�p�1

�r

; (2-17)

that is,

F1.2� ip�1z; �/D

1

2�p�1

E1.z; �/ ˜

REMARK 2.11. The algebra of quasi-Jacobi formsCŒe2;E1;E2; : : : � is closedunder differentiation with respect to� and@z. Indeed, one has

2� i@E1

@�DE3�E1E2;

@E1

@zD�E2;

2� i@E2

@�D 3E4� 2E1E3�E2

2 ;@E2

@zD�2E3;

and henceCŒ: : : ;Ei ; : : :� is a D-module, whereD is the ring of differentialoperators generated by@=@� and @=@z over the ring of holomorphic Jacobigroup invariant functions onH �C. As is clear from the discussion, the ring ofEisenstein seriesCŒ: : : ;Ei ; : : :� has a natural identification with the ring of realvalued almost meromorphic Jacobi formsCŒE�

1;E�

2;E3; : : : � on H�C having

index zero, where

E�

1 DE1C 2� iIm x

Im �; E�

2 DE2C1

Im �: (2-18)

THEOREM 2.12. The algebra of quasi-Jacobi forms of depth.k; 0/, k � 0, isisomorphic to the algebra of complex unitary cobordisms modulo flops.

In another direction, the depth of quasi-Jacobi forms allows one to “measure”the deviation of the elliptic genus of a non-Calabi–Yau manifold from being aJacobi form.

THEOREM 2.13. Elliptic genera of manifolds of dimension at mostd span thesubspace of forms of depth.d; 0/ in the algebra of quasi-Jacobi forms. If acomplex manifold satisfiesck

1D 0 andck�1

1¤ 0, 1 its elliptic genus is a quasi-

Jacobi form of depth.s; 0/, wheres � k � 1.

1More generally,k is the smallest among indicesi with ci1

2 Ann.c2; : : : ; cdimM /; an example of sucha manifold is ann-manifold having a.n�k/-dimensional Calabi–Yau factor.


PROOF. It follows from the proof of Proposition (2.10) that

d2 log˚

dx2DX

i�2

.�1/i�1 i � 1

.2�p�1/i

.Ei � ei/xi�2;

which yields

˚ D eE1xY

i

e.1=i/.�1/i�1.i�1/=.2�p

�1/i .Ei �ei /xi

: (2-19)

The Hirzebruch characteristic series is

˚� x

2� i

� �.z/

� 0.0/I

compare (1-1). Hence, ifc.TX /D˘.1Cxk/, then

El l.X /D

�

�.z/

� 0.0/

�dimXY

i;k

eE1xk e.1=i/.�1/i�1.i�1/.Ei �ei /xik ŒX �

D

�

�.z/

� 0.0/

�dimX

ec1.X /E1

Y

i;k

e.1=i/.�1/i�1.i�1/.Ei �ei /xik ŒX �; (2-20)

whereŒX � is the fundamental class ofX . In other words, ifc1 D 0, the ellipticclass is a polynomial inEi � ei with i � 2, and hence the elliptic genus is aJacobi form [36]. Moreover ofck

1D 0 the degree of this polynomial is at most

k in E1, and the claim follows. ˜

EXAMPLE 2.14. Expression (2-20) can be used to get formulas for the ellipticgenus of specific examples in terms of Eisenstein seriesEn. For example, for asurface inP3 having degreed one has

�

E21.

12d2� 4d C 8/d C .E2� e2/.

12d2� 2/d

�

�

�.z/

� 0.0/

�2

In particular ford D 1 one obtains

�

92E2

1� 3

2.E2� e2/

�

�

�.z/

� 0.0/

�2

:

One can compare this with the double series that is a special case of the generalformula for the elliptic genus of toric varieties in [6]. This leads to a two-variableversion of the identity discussed in [6, Remark 5.9]. In factfollowing [5] onecan define the subalgebra of “toric quasi-Jacobi forms” of the algebra of quasi-Jacobi forms, extending the toric quasimodular forms considered in [5]. Thisissue will be addressed elsewhere.


Next we consider one more similarity between meromorphic Jacobi forms andmodular forms: there is a natural noncommutative deformation of the ordinaryproduct of Jacobi forms similar to the deformation of the product modular formsconstructed using Rankin–Cohen brackets [57]. In fact we have the followingJacobi counterpart of Rankin–Cohen brackets:

PROPOSITION2.15. Letf andg be Jacobi forms of index zero and weightsk

andl , respectively. Then

Œf;g�D k�

@�f �1

2� iE1@zf

�

g� l�

@�g�1

2� iE1.z; �/@zg

�

f

is a Jacobi form of weightkC l C 2. More generally, let

D D @� �1

2� iE1@z :

Then the Cohen–Kuznetsov series(see [57])

QfD.z; �;X /D

1X

nD0

Dnf .z; �/X n

n!.k/n;

where.k/n D k.kC 1/ � � � .kC n� 1/ is the Pochhammer symbol, satisfies

QfD

�

a� C b

c� C d;

z

c� C d;

z

c� C d;

X

.c� C d/2

�

D .c� C d/k exp

�

c

c� C d

X

2� i

�

fD.�; z;X /;

QfD.�; zC a� C b;X /D QfD.�; z;X /:

In particular, the coefficientŒf;g�n=.k/n.l/n of X n in QfD.�; z;�X / QgD.�; z;X /

is a Jacobi form of weightkC l C 2n. It is given explicitly in terms ofDif andDj g by the same formulas as the classical RC brackets.

PROOF. The main point is that the operator@� � 12� i

E1@z has the same devia-tion from transforming a Jacobi form into another as@� has on modular forms.Indeed:

.@� �1

2� iE1@z/f

�

a�Cb

c�Cd;

z

c�Cd

�

D�

kc.c�Cd/kC1f .�; z/Czc.c�Cd/kC1@zf .�; z/C.c�Cd/kC2@�f .�; z/

�1

2� i

�

.c� C d/E1.�; z/C 2� icz�

.c� C d/kC1@zf�

D .c� C d/kC2�

f .�; z/�1

2� iE1fz

�

C kc.c� C d/kC1f .�; z/:


Moreover,�

@� �1

2� iE1@z

�

f .�; zC a� C b/D f� C afz �1

2� i.E1� 2� ia/fz

D�

@� �1

2� iE1@z

�

f .�; z/:

The rest of the proof runs as in [57]. ˜

REMARK 2.16. The brackets introduced in Proposition 2.15 are different fromthe Rankin–Cohen bracket introduced in [13].

3. Real singular varieties

The Ochanine genus of an oriented differentiable manifoldX can be definedusing the following series with coefficients inQŒŒq�� as the Hirzebruch charac-teristic power series (see [37] and references there):

Q.x/Dx=2

sinh.x=2/

1Y

nD1

�

.1� qn/2

.1� qnex/.1� qne�x/

�.�1/n

(3-1)

As was mentioned in Section 1B, this genus is a specialization of the two-variable elliptic genus (atz D 1

2). Evaluation of the Ochanine genus of a mani-

fold using (3-1) and viewing the result as function of� on the upper half-plane(whereq D e2� i� ) yields a modular form on�0.2/� SL2.Z/; see [37].

In this section we discuss elliptic genera for real algebraic varieties. It par-ticular we address Totaro’s proposal [48] that “it should bepossible to defineOchanine genus for a large class of compact oriented real analytic spaces.” Inthis direction we have:

THEOREM 3.1 [48]. The quotient ofMSO by the ideal generated by orientedreal flops and complex flops(that is, the ideal generated byX 0 �X , whereX 0

andX are related by a real or complex flop) is

Z Œı; 2 ; 2 2; 2 4�;

with CP2 corresponding toı andCP4 to 2 C ı2. This quotient ring is the theimage ofMSO� under the Ochanine genus.

In particular the Ochanine genus of a small resolution is independent of itschoice for singular spaces having singularities only alongnonsingular strata andhaving in normal directions only singularities which are cones inR4 or C4.

Our goal is to find a wider class of singular real algebraic varieties for whichthe Ochanine genus of a resolution is independent of the choice of the latter.


3A. Real singularities. For the remainder of this paper “real algebraic variety”means anoriented quasiprojective varietyXR over R, X.R/ is the set of itsR-points with the Euclidean topology,XCDXR�SpecR SpecC is the complexi-fication andX.C/ the analytic space of its complex points. We also assume thatdimR X.R/D dimC X.C/.

DEFINITION 3.2. A real algebraic varietyXR as above is calledQ-Gorensteinlog-terminal if the analytic spaceX.C/ is Q-Gorenstein log-terminal.

EXAMPLE 3.3. The affine variety

x21 �x2

2 Cx23 �x2

4 D 0 (3-2)

in R4 is three-dimensional Gorenstein log-terminal and admits acrepant reso-lution.

Indeed, it is well known that the complexification of the Gorenstein singularity(3-2) admits a small (and hence crepant) resolution havingP1 as its exceptionalset.

EXAMPLE 3.4. The three-dimensional complex cone inC4 given byz21C z2

2C

z23Cz2

4D0 considered as a codimension-two subvariety ofR8 is aQ-Gorenstein

log-terminal variety overR and its complexification admits a crepant resolution.

Indeed, this codimension-two subvariety is a real analyticspace which is theintersection of two quadrics inR8 given by

a21Ca2

2Ca23Ca2

4�b21�b2

2�b23�b2

4 D 0D a1b1Ca2b2Ca3b3Ca4b4; (3-3)

whereai D Rezi ; bi D Im zi . The complexification is the cone over completeintersection of two quadrics inP7. Moreover, the defining equations of thiscomplete intersection become, after the change of coordinatesxiDaiC

p�1bi ,

yi D ai �p�1bi ,

x21 Cx2

2 Cx23 Cx2

4 D 0 and y21 Cy2

2 Cy23 Cy2

4 D 0: (3-4)

The singular locus is the union of two disjoint two-dimensional quadrics and thesingularity along each isA1 (i.e., the intersection of the transversal to it inP7

has anA1 singularity). To resolve (3-3), one can blow upC8 at the origin, whichresults in aC-fibration over the complete intersection (3-4). It can be resolvedby small resolutions along two nonsingular components of the singular locus of(3-4). A direct calculation (considering, for example, theorder of the pole of theform dx2 ^ dx3 ^ dx4 ^ dy2 ^ dy3 ^ dy4=.x1y1/ along the intersection of theexceptional locus of the blow-upQC8 of C8 with the proper preimage of (3-4) inQC8) shows that we have a log-terminal resolution of the Gorenstein singularity

which is the complexification of (3-3).


3B. The elliptic genus of resolutions of real varieties withQ-Gorensteinlog-terminal singularities. Let X be a real algebraic manifold and letD DP

˛kDk , with each˛k in Q, be a divisor on the complexificationXC of X

(i.e., theDk are irreducible components ofD). Let xi denote the Chern rootsof the tangent bundle ofXC and denote bydk the classes corresponding toDk

(Section 1).

DEFINITION 3.5. LetX be a real algebraic manifold andD a divisor on thecomplexificationXC of X . The Ochanine classELLO.X;D/ of the pair.X;D/is the specialization

ELL�

XC;D; q; z D12

�

of the two-variable elliptic class of the pairELL.XC;D; q; z/ given by

�

Y

l

�

xl

2� i

�

��

xl

2� i � z�

� 0.0/

�.�z/��

xl

2� i

�

�

�

�

Y

k

��

dk

2� i � .˛k C 1/z�

�.�z/

��

dk

2� i � z�

�.�.˛k C 1/z/

�

: (3-5)

The Ochanine elliptic genus of the pair.X;D/ as above is

Ell.XR;D/D

q

ELL�

XC;D; q;12

�

[ cl.X.R//ŒX.C/�: (3-6)

HerepELL denotes the class corresponding to the unique series with constant

term 1 and havingELL as its square.The class of pairs above is the class (1-3) considered in Definition 1.4 in the

case where groupG is trivial. One can define an orbifold version of this classas well, specializing (1-3) toz D 1

2. See [8] for further discussion of the class

ELL.X;D/.The relation with Ochanine’s definition is as follows: ifD is the trivial divisor

on XC, the result coincides with the genus [43]. More precisely:

LEMMA 3.6. LetXR be a real algebraic manifold with nonsingular complexifi-cationXC. Then

Ell.XR/Dq

ELL.TX .C//[ cl.X.R//ŒX.C/�:

PROOF. Indeed, we have

0! TX .R/! TX .C/jX .R/! TX .R/! 0; (3-7)

with the identification of the normal bundle toXR with its tangent bundle givenby multiplication by

p�1. HenceELL.XR/

2 D i�.ELLXC/, wherei W XR!

XC is the canonical embedding. Now the lemma follows from the identifica-tion of the characteristic series (3-1) and specializationz D 1

2of the series in

(1-1) (see [6]) and the identification which is just a definition of the class clZ 2


H dimR Y �dimRZ of a submanifoldZ of a manifoldY : clZ [˛ŒY �D i�.˛/\ ŒZ�

for any˛ 2H dimR Z .Y /. Indeed, we have

Ell.XR/D ELL.TX .R//ŒX.R/�Dq

ELL.TX .C//ˇ

ˇ

X .R/ŒX.R/�

Dq

ELL.TX .C//[ cl.X.R//ŒX.C/�: ˜

Our main result in this section is the following:

THEOREM 3.7. Let � W . QX ; QD/ ! .X;D/ be a resolution of singularities ofa real algebraic pair withQ-Gorenstein log-terminal singularities; i.e., K QX

C

QD D ��.KX CD/. Then the elliptic genus of the pair. QX ; QD/ is independentof the resolution. In particular, if a real algebraic varietyX has a crepantresolution, its elliptic genus is independent of a choice of crepant resolution.

PROOF. Indeed for a blowupf W . QX ; QD/! .X;D/ we have

f�

�

q

ELL�

QX ; QD; q; 12

�

�

D

q

ELL�

X;D; q; 12

�

(3-8)

This is a special case of the push-forward formula (1-4) in theorem 1.5, withGbeing the trivial group. Hence

ELLO.XR;D/D

q

ELL�

XC;D; q;12

�

[ cl.XR/ŒXC�

D

q

ELL. QXC; QD; q;12

�

[f �.ŒXR�\ ŒXC�/D ELL. QXR; QD/;

as follows from projection formula sincef �.cl ŒXR�/D Œcl QXR� and sincef� isthe identity onH0.

For a crepant resolution one hasD D 0 and hence by Lemma 3.6 the ellipticgenus ofXR is the Ochanine genus of the real manifold, which is its crepantresolution. ˜

REMARK 3.8. Examples 3.3 and 3.4 show that singularities admittinga crepantresolution include real three-dimensional cones and real points of complex three-dimensional cones.

Acknowledgement

The author wants to express his gratitude to Lev Borisov. Thematerial inSection 1 is a survey of joint papers with him, and the resultsof Section 2 arebased on discussions with him several years ago.


References

[1] D. Abramovich, K. Karu, K. Matsuki, J. Włodarsczyk, Torification and factorizationof birational maps. J. Amer. Math. Soc., 15 (2002) no. 3, 531–572.

[2] V. Batyrev, Non-archimedean integrals and stringy Euler numbers of log-terminalpairs. J. Eur. Math. Soc. 1 (1999), no. 1, 5–33.

[3] J. Block, S. Weinberger, Higher Todd classes and holomorphic group actions. PureAppl. Math. Q. 2 (2006), no. 4:2, 1237–1253.

[4] R. Borcherds, Automorphic forms on OsC2;2.R/C and generalized Kac–Moodyalgebras. In Proceedings of the International Congress of Mathematicians (Zurich,1994), 744–752, Birkhauser, Basel, 1995.

[5] L. Borisov, P. Gunnells, Toric modular forms, Inventiones Math. 144 (2001), no. 2,297–325.

[6] L. Borisov, A. Libgober, Elliptic genera of toric varieties and applications to mirrorsymmetry. Invent. Math. 140 (2000), no. 2, 453–485.

[7] L. Borisov, A. Libgober, Elliptic genera of singular varieties, orbifold elliptic genusand chiral de Rham complex. In Mirror symmetry, IV (Montreal, QC, 2000), 325–342, AMS/IP Stud. Adv. Math., 33, Amer. Math. Soc., Providence, RI, 2002.

[8] L. Borisov, A. Libgober, Elliptic genera of singular varieties. Duke Math. J. 116(2003), no. 2, 319–351.

[9] L. Borisov, A. Libgober, McKay correspondence for elliptic genera. Ann. of Math.(2) 161 (2005), no. 3, 1521–1569.

[10] L. Borisov, A. Libgober, Higher elliptic genera. Math.Res. Lett. 15 (2008), no. 3,511–520.

[11] S. Cappell, A. Libgober, L. Maxim, J. Shaneson, Hodge genera and characteristicclasses of complex algebraic varieties. Electron. Res. Announc. Math. Sci. 15 (2008),1–7.

[12] K. ChandrasekharanElliptic functions, Grundlehren Math. Wiss.281, Springer,Berlin, 1985.

[13] Y. J. Choie, W. Eholzer, Rankin–Cohen operators for Jacobi and Siegel forms. J.Number Theory 68 (1998), no. 2, 160–177.

[14] I. Ciocan-Fontanine, M. Kapranov, Virtual fundamental classes via dg-manifolds,arXiv math/0703214.

[15] J. Davis,Manifold aspects of the Novikov conjecture, in Surveys in surgery theory,vol. 1, 195–224, Ann. Math. Studies145, Princeton Univ. Press, Princeton, 2000.

[16] P. Deligne, Theorie de Hodge II, Publ. Math. IHES 40 (1971), 5–58.

[17] P. Deligne, Theorie de Hodge III, Publ. Math. IHES 44 (1974), 5–77.

[18] R. Dijkgraaf, D. Moore, E. Verlinde, H. Verlinde,Elliptic genera of symmetricproducts and second quantized strings. Comm. Math. Phys.185(1997), no. 1, 197–209.


[19] J. Denef, F. Loeser, Geometry on arc spaces of algebraicvarieties. In Proceedingsof European Congress of Mathematics, vol. I (Barcelona, 2000), 327–348, Progr.Math.201, Birkhauser, Basel, 2001.

[20] C. Dong, K. F. Liu, X. Ma, On orbifold elliptic genus. In Orbifolds in mathematicsand physics (Madison, WI, 2001), 87–105, Contemp. Math.310, Amer. Math. Soc.,Providence, RI, 2002.

[21] B. Fantechi, L. Gottsche Riemann-Roch theorems and elliptic genus for virtuallysmooth Schemes. arXiv 0706.0988.

[22] E. Frenkel, M. Szczesny, Chiral de Rham complex and orbifolds. J. AlgebraicGeom. 16 (2007), no. 4, 599–624.

[23] W. Fulton, Intersection Theory (second edition), Springer, Berlin, 1998.

[24] M. Gaberdiel, C. Keller, Differential operators for elliptic genera. arXiv0904.1831.

[25] E. Gasparim, Melissa Liu, The Nekrasov conjecture for toric surfaces, arXiv0808.0884

[26] M. Goresky, R. MacPherson. Problems and bibliography on intersection homol-ogy. In Intersection cohomology, edited by A. Borel et al., 221–233. Birkhauser,Boston, 1984.

[27] V. Gorbounov,S. Ochanine, Mirror symmetry formulae for the elliptic genus ofcomplete intersections. J. Topol. 1 (2008), no. 2, 429–445.

[28] F. Hirzebruch, Topological methods in algebraic geometry (reprint of the 1978edition), Classics in Mathematics, Springer, Berlin, 1995.

[29] F. Hirzebruch, Elliptic genera of levelN for complex manifolds. In Differentialgeometric methods in theoretical physics (Como, 1987), edited by K. Bleuer andM. Werner, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci.250. Dordrecht, Kluwer,1988.

[30] F. Hirzebruch, T. Berger, R. Jung, Manifolds and modular forms. Aspects ofMathematicsE20. Vieweg, Braunschweig, 1992.

[31] M. Kaneko, D. Zagier, A generalized Jacobi theta function and quasimodularforms. In The moduli space of curves (Texel Island, 1994), 165–172, Progr. Math.129, Birkhauser, Boston, 1995.

[32] M. Kapranov, E. Vasserot, Vertex algebras and the formal loop space. Publ. Math.IHES 100 (2004), 209–269.

[33] M. Kapranov, E. Vasserot, Formal loops IV: Chiral differential operators, arXivmath/0612371.

[34] T. Kawai, Y. Yamada, and S.-K. Yang, Elliptic genera andN D 2 superconformalfield theory, Nuclear Phys. B 414 (1994), no. 1–2, 191–212.

[35] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model prob-lem. In Algebraic geometry (Sendai, 1985), 283–360, Adv. Stud. Pure Math.10,North-Holland, Amsterdam, 1987.


[36] I. Krichever, Generalized elliptic genera and Baker–Akhiezer functions, Math.Notes, 47 (1990), no. 2, 132–142.

[37] Elliptic curves and modular forms in algebraic topology, edited by P. S. Landwe-ber. Springer, Berlin, 1988.

[38] J. Li, K. F. Liu and J. Zhou. Topological string partition functions and equivariantindices, Asian J. Math. 10 (2006), no. 1, 81–114.

[39] A. Libgober, M. Szczesny, Discrete torsion, orbifold elliptic genera and the chiralde Rham complex. Pure Appl. Math. Q. 2 (2006), no. 4:2, 1217–1236.

[40] E. Looijenga, Motivic measures. In Seminaire Bourbaki, 1999/2000. Asterisque276(2002), 267–297.

[41] F. Malikov, V. Schechtman, A. Vaintrob, Chiral de Rham complex. Comm. Math.Phys. 204 (1999), no. 2, 439–473.

[42] F. Malikov, V. Gorbounov. Vertex algebras and the Landau–Ginzburg/Calabi–Yaucorrespondence. Mosc. Math. J. 4 (2004), no. 3, 729–779.

[43] S. Ochanine, Sur les genres multiplicatifs definis par des integrales elliptiques.Topology 26 (1987), no. 2, 143–151.

[44] J. Rosenberg, An analogue of the Novikov conjecture in complex algebraic geom-etry. Trans. Amer. Math. Soc. 360 (2008), no. 1, 383–394.

[45] J. Tannery, J. Molk, Elements de la theorie des fonctions elliptiques. Chelsea,Bronx, NY, 1972.

[46] C. B. Thomas, Elliptic cohomology. Kluwer/Plenum, NewYork, 1999.

[47] B. Totaro, Chern numbers for singular varieties and elliptic homology. Ann. Math.(2) 151 (2000), no. 2, 757–791.

[48] B. Totaro. The elliptic genus of a singular variety. In Elliptic cohomology, 360–364, London Math. Soc. Lecture Note Ser.342, Cambridge Univ. Press, Cambridge,2007.

[49] R. Waelder, Equivariant elliptic genera, Pacific Math.J. 235 (2008), no. 2, 345-277.

[50] R. Waelder, Singular McKay correspondence for normal surfaces. arXiv0810.3634

[51] A. Weil, Elliptic functions according to Eisenstein and Kronecker. Springer,Berlin, 1999.

[52] C. L. Wang,K-equivalence in birational geometry and characterizations of com-plex elliptic genera. J. Algebraic Geom. 12 (2003), no. 2, 285–306.

[53] E. Witten, The index of the Dirac operator in loop space.In Elliptic curves andmodular forms in algebraic topology (Princeton, 1986), 161–181 edited by P. S.Landweber, Lecture Notes in Math.1326, Springer, New York, 1988.

[54] E. Witten, Mirror manifolds and topological field theory. In Essays on mirrormanifolds. edited by Shing-Tung Yau, 120–158. International Press, Hong Kong,1992.


[55] D. Zagier, Note on the Landweber-Stong elliptic genus.In Elliptic curves andmodular forms in algebraic topology (Princeton, 1986), edited by P. S. Landweber,216–224, Lecture Notes in Math.1326, Springer, New York, 1988.

[56] D. Zagier, Periods of modular forms and Jacobi theta functions. Invent. Math. 104(1991), no. 3, 449–465.

[57] D. Zagier, Elliptic modular forms and their applications. In The 1-2-3 of modularforms, edited by Kristian Ranestad, 1–103, Universitext, Springer, Berlin, 2008.

ANATOLY L IBGOBER


UNIVERSITY OF ILLINOIS

CHICAGO, IL [email protected]


The weight filtration for real algebraic varietiesCLINT MCCRORY AND ADAM PARUSINSKI

ABSTRACT. Using the work of Guillen and Navarro Aznar we associate toeach real algebraic variety a filtered chain complex, the weight complex, whichis well-defined up to a filtered quasi-isomorphism, and induces on Borel–Moore homology withZ2 coefficients an analog of the weight filtration forcomplex algebraic varieties.

The weight complex can be represented by a geometrically defined filtrationon the complex of semialgebraic chains. To show this we definethe weightcomplex for Nash manifolds and, more generally, for arc-symmetric sets, andwe adapt to Nash manifolds the theorem of Mikhalkin that two compact con-nected smooth manifolds of the same dimension can be connected by a se-quence of smooth blowups and blowdowns.

The weight complex is acyclic for smooth blowups and additive for closedinclusions. As a corollary we obtain a new construction of the virtual Bettinumbers, which are additive invariants of real algebraic varieties, and we showtheir invariance by a large class of mappings that includes regular homeomor-phisms and Nash diffeomorphisms.

The weight filtration of the homology of a real variety was introduced byTotaro [37]. He used the work of Guillen and Navarro Aznar [15] to showthe existence of such a filtration, by analogy with Deligne’sweight filtrationfor complex varieties [10] as generalized by Gillet and Soule [14]. There is alsoearlier unpublished work on the real weight filtration by M. Wodzicki, and morerecent unpublished work on weight filtrations by Guillen and Navarro Aznar[16].

Totaro’s weight filtration for a compact variety is associated to the spectralsequence of a cubical hyperresolution. (For an introduction to cubical hyper-resolutions of complex varieties see [34], Chapter 5.) For complex varieties

Mathematics Subject Classification:Primary: 14P25. Secondary: 14P10, 14P20.

Research partially supported by a grant from Mathematiques en Pays de la Loire (MATPYL)..

121

122 CLINT MCCRORY AND ADAM PARUSINSKI

this spectral sequence collapses with rational coefficients, but for real varieties,where it is defined withZ2 coefficients, the spectral sequence does not collapsein general. We show, again using the work of Guillen and Navarro Aznar, thatthe weight spectral sequence is itself a natural invariant of a real variety. There isa functor that assigns to each real algebraic variety a filtered chain complex, theweight complex, that is unique up to filtered quasi-isomorphism, and functorialfor proper regular morphisms. The weight spectral sequenceis the spectralsequence associated to this filtered complex, and the weightfiltration is thecorresponding filtration of Borel–Moore homology with coefficients inZ2.

Using the theory of Nash constructible functions we give an independentconstruction of a functorial filtration on the complex of semialgebraic chainsin Kurdyka’s category of arc-symmetric sets [19; 21], and weshow that thefiltered complex obtained in this way represents the weight complex of a realalgebraic variety. We obtain in particular that the weight complex is invariantunder regular rational homeomorphisms of real algebraic sets in the sense ofBochnak, Coste and Roy [5].

The characteristic properties of the weight complex describe how it behaveswith respect to generalized blowups (acyclicity) and inclusions of open subvari-eties (additivity). The initial term of the weight spectralsequence yields additiveinvariants for real algebraic varieties, the virtual Bettinumbers [24]. Thus weobtain that the virtual Betti numbers are invariants of regular homeomorphismsof real algebraic sets. For real toric varieties, the weightspectral sequence isisomorphic to the toric spectral sequence introduced by Bihan, Franz, McCrory,and van Hamel [4].

In Section 1 we prove the existence and uniqueness of the filtered weightcomplex of a real algebraic variety. The weight complex is the unique acyclicadditive extension to all varieties of the functor that assigns to a nonsingular pro-jective variety the complex of semialgebraic chains with the canonical filtration.To apply the extension theorems of Guillen and Navarro Aznar [15], we work inthe category of schemes overR, for which one has resolution of singularities, theChow–Hironaka Lemma (see [15, (2.1.3)]), and the compactification theorem ofNagata [28]. We obtain the weight complex as a functor of schemes and properregular morphisms.

In Section 2 we characterize the weight filtration of the semialgebraic chaincomplex using resolution of singularities. In Section 3 we introduce the Nashconstructible filtration of semialgebraic chains, following Pennaneac’h [32], andwe show that it gives the weight filtration. A key tool is Mikhalkin’s theorem[26] that any two connected closedC 1 manifolds of the same dimension canbe connected by a sequence of blowups and blowdowns. Section4 we presentseveral applications to real geometry.

THE WEIGHT FILTRATION FOR REAL ALGEBRAIC VARIETIES 123

In Section 5 we show that for a real toric variety the Nash constructible filtra-tion is the same as the filtration on cellular chains defined byBihanet al.usingtoric topology.

1. The homological weight filtration

We begin with a brief discussion of the extension theorem of Guill en andNavarro Aznar. Suppose thatG is a functor defined for smooth varieties overa field of characteristic zero. The main theorem of [15] givesa criterion forthe extension ofG to a functorG0 defined for all (possibly singular) varieties.This criterion is a relation between the value ofG on a smooth varietyX andthe value ofG on the blowup ofX along a smooth center. The extensionG0

satisfies a generalization of this blowup formula for any morphismf W QX !X

of varieties that is an isomorphism over the complement of a subvarietyY of X .If one requires an even stronger additivity formula forG0.X / in terms ofG0.Y /

andG0.X nY /, then one can assume that the original functorG is defined onlyfor smooth projective varieties.

The structure of the target category of the functorG is important in this theory.The prototype is the derived category of chain complexes in an abelian category.That is, the objects are chain complexes, and the set of morphisms betweentwo complexes is expanded to include the inverses of quasi-isomorphisms (mor-phisms that induce isomorphisms on homology). Guillen and Navarro introducea generalization of the category of chain complexes called adescent category,which has a class of morphismsE that are analogous to quasi-isomorphisms,and a functors from diagrams to objects that is analogous to the total complexof a diagram of chain complexes.

In our application we consider varieties over the field of real numbers, andthe target category is the derived category of filtered chaincomplexes of vectorspaces overZ2. Since this category is closely related to the classical categoryof chain complexes, it is not hard to check that it is a descentcategory. Ourstarting functorG is rather simple: It assigns to a smooth projective varietythe complex of semialgebraic chains with the canonical filtration. The blowupformula follows from a short exact sequence (1-3) for the homology groups ofa blowup.

Now we turn to a precise statement and proof of Theorem 1.1, which is ourmain result.

By a real algebraic varietywe mean a reduced separated scheme of finitetype overR. By acompactvariety we mean a scheme that is complete (properover R). We adopt the following notation of Guillen and Navarro Aznar [15].Let Schc.R/ be the category of real algebraic varieties and proper regular mor-phisms,i. e. proper morphisms of schemes. By Reg we denote the subcategory


of compact nonsingular varieties, and byV.R/ the category of projective non-singular varieties. A proper morphism or a compactificationof varieties willalways be understood in the scheme-theoretic sense.

In this paper we are interested in the topology of the set of real points of areal algebraic varietyX . Let X denote the set of real points ofX . The setX , with its sheaf of regular functions, is a real algebraic variety in the sense ofBochnak, Coste and Roy [5]. For a varietyX we denote byC�.X / the complexof semialgebraic chains ofX with coefficients inZ2 and closed supports. Thehomology ofC�.X / is the Borel–Moore homology ofX with Z2 coefficients,and will be denoted byH�.X /.

1A. Filtered complexes. Let C be the category of bounded complexes ofZ2

vector spaces with increasing bounded filtration,

K�D� � � K0 K1 K2 � � � ; � � ��Fp�1K��FpK��FpC1K�� :

Such a filtered complex defines a spectral sequencefEr ; dr g, r D 1; 2; : : : , with

E0p;q D

FpKpCq

Fp�1KpCq; E1

p;q DHpCq

�

FpK�

Fp�1K�

�

;

that converges to the homology ofK�,

E1

p;q DFp.HpCqK�/

Fp�1.HpCqK�/;

whereFp.HnK�/ D ImageŒHn.FpK�/ ! Hn.K�/�; see [22, Thm. 3.1]. Aquasi-isomorphismin C is a filtered quasi-isomorphism, that is, a morphism offiltered complexes that induces an isomorphism onE1. Thus a quasi-isomo-phism induces an isomorphism of the associated spectral sequences.

Following (1.5.1) in [15], we denote by HoC the categoryC localized withrespect to filtered quasi-isomorphisms.

Every bounded complexK� has acanonical filtration[8] given by

F canp K� D

(

Kq if q > �p,ker@q if q D�p,0 if q < �p.

We have

E1p;q DHpCq

F canp K�

F canp�1

K�

!

Dn

HpCq.K�/ if pC q D�p,0 otherwise.

(1-1)

Thus a quasi-isomorphism of complexes induces a filtered quasi-isomorphismof complexes with canonical filtration.

To certain types of diagrams inC we can associate an element ofC, thesimplefiltered complexof the given diagram. We use notation from [15]. Forn� 0 let


˜Cn be the partially ordered set of subsets off0; 1; : : : ; ng. A cubical diagram

of type˜Cn in a categoryX is a contravariant functor from C

n to X . If K is acubical diagram inC of type˜

Cn , let K�;S be the complex labeled by the subset

S � f0; 1; : : : ng, and letjS j denote the number of elements ofS . The simplecomplex sK is defined by

sKk DM

iCjS j�1Dk

Ki;S

with differentials@ W sKk ! sKk�1 defined as follows. For eachS let @0 W

Ki;S ! Ki�1;S be the differential ofK�;S . If T � S and jT j D jS j � 1, let@T;S W K�;S !K�;T be the chain map corresponding to the inclusion ofT inS . If a 2Ki;S , let

@00.a/DX

@T;S .a/;

where the sum is over allT � S such thatjT j D jS j � 1, and

@.a/D @0.a/C @00.a/:

The filtration of sK is given byFp sKD sFpK,

.Fp sK/k DM

iCjS j�1Dk

Fp.Ki;S /:

The simple complex functors is defined for cubical diagrams in the categoryC, but not for diagrams in the derived category HoC, since a diagram in HoCdoes not necessarily correspond to a diagram inC. However, for eachn � 0,the functor s is defined on the derived category of cubical diagrams of type˜

Cn . (A quasi-isomorphism in the category of cubical diagrams of type ˜

Cn

is a morphism of diagrams that is a quasi-isomorphism on eachobject in thediagram.)

To address this technical problem, Guillen and Navarro Aznar introduce the˚-rectificationof a functor with values in a derived category [15, (1.6)], where˚ is the category of finite orderable diagrams [15, (1.1.2)]. A(˚-)rectificationof a functorG with values in a derived category HoC is an extension ofG to afunctor of diagrams, with values in the derived category of diagrams, satisfyingcertain naturality properties [15, (1.6.5)]. A factorization of G through the cat-egoryC determines a canonical rectification ofG. One says thatG is rectifiedif a rectification ofG is given.


1B. The weight complex.To state the next theorem, we only need to considerdiagrams in of type C

0or type˜

C

1. The inclusion of a closed subvarietyY �X

is a˜C

0-diagram inSchc.R/. An acyclic square([15], (2.1.1)) is a C

1-diagram

in Schc.R/,

QY ��! QX?

?

y

?

?

y

�

Yi

��! X

(1-2)

wherei is the inclusion of a closed subvariety,QY D ��1.Y /, and the restrictionof � is an isomorphismQX n QY ! X n Y . An elementary acyclic squareis anacyclic square such thatX is compact and nonsingular,Y is nonsingular, and�is the blowup ofX alongY .

For a real algebraic varietyX , let F canC�.X / denote the complexC�.X / ofsemialgebraic chains with the canonical filtration.

THEOREM 1.1. The functor

F canC� W V.R/! HoC

that associates to a nonsingular projective varietyM the semialgebraic chaincomplex ofM with canonical filtration admits an extension to a functor definedfor all real algebraic varieties and proper regular morphisms,

WC � W Schc.R/! HoC;

such thatWC � is rectified and has the following properties:

(i) Acyclicity. For an acyclic square(1-2) the simple filtered complex of thediagram

WC �. QY / ��! WC �. QX /?

?

y

?

?

y

WC �.Y / ��! WC �.X /

is acyclic(quasi-isomorphic to the zero complex).(ii) Additivity. For a closed inclusionY �X , the simple filtered complex of the

diagram

WC �.Y /!WC �.X /

is naturally quasi-isomorphic toWC �.X nY /.

Such a functorWC � is unique up to a unique quasi-isomorphism.


PROOF. This theorem follows from [15], Theorem.2:2:2/op. By Proposition.1:7:5/op of [15], the categoryC, with the class of quasi-isomorphisms and theoperation of simple complexs defined above, is a category of homologicaldescent. Since it factors throughC, the functorF canC� is ˚-rectified ([15],(1.6.5), (1.1.2)). ClearlyF canC� is additive for disjoint unions (condition (F1)of [15]). It remains to check condition (F2) forF canC�, that the simple filteredcomplex associated to an elementary acyclic square is acyclic.

Consider the elementary acyclic square (1-2). LetK be the simple complexassociated to the C

1-diagram

F canC�. QY / ��! F canC�. QX /?

?

y

?

?

y

F canC�.Y / ��! F canC�.X /

By definition of the canonical filtration, for eachp we have

.Fp sK/k=.Fp�1 sK/k ¤ 0 only for �pC 2� k � �p� 1;

and the complex.E0p;�; d

0/ has the form

0!.Fp sK/�pC2

.Fp�1 sK/�pC2

!.Fp sK/�pC1

.Fp�1 sK/�pC1

!.Fp sK/�p

.Fp�1 sK/�p!

.Fp sK/�p�1

.Fp�1 sK/�p�1

! 0:

A computation gives

H�pC2.E0p;�/D 0;

H�pC1.E0p;�/D KerŒH�p. QY /!H�p.Y /˚H�p. QX /�;

H�p.E0p;�/D

KerŒH�p.Y /˚H�p. QX /!H�p.X /�=ImŒH�p. QY /!H�p.Y /˚H�p. QX /�;

H�p�1.E0p;�/DH�p.X /=ImŒH�p.Y /˚H�p. QX /!H�p.X /�:

These groups are zero because for allk we have the short exact sequence of anelementary acyclic square,

0!Hk. QY /!Hk.Y /˚Hk. QX /!Hk.X /! 0I (1-3)

see [25], proof of Proposition 2.1. ˜

REMARK 1.2. This above argument shows that the functorF can is acyclic onany acyclic square (1-2), provided the varietiesX;Y; QX ; QY are nonsingular andcompact.


REMARK 1.3. In Section 3 below, we show that the functorWC � factorsthrough the category of filtered chain complexes. This explains whyWC � isrectified.

If X is a real algebraic variety, theweight complexof X is the filtered com-plex WC�.X /. A stronger version of the uniqueness ofWC � is given by thefollowing naturality theorem.

THEOREM1.4. LetA�, B� WV.R/!C be functors whose localizationsV.R/!HoC satisfy the disjoint additivity condition(F1) and the elementary acyclicitycondition (F2) of [15]. If � W A� ! B� is a morphism of functors, then thelocalization of� extends uniquely to a morphism� 0 WWA�!WB�.

PROOF. This follows from.2:1:5/op and.2:2:2/op of [15]. ˜

Thus if � W A�.M /! B�.M / is a quasi-isomorphism for all nonsingular pro-jective varietiesM , then� 0 WWA�.X /!WB�.X / is a quasi-isomorphism forall varietiesX .

PROPOSITION 1.5. For all real algebraic varietiesX , the homology of thecomplexWC�.X / is the Borel–Moore homology ofX with Z2 coefficients,

Hn.WC�.X //DHn.X /:

PROOF. Let D be the category of bounded complexes ofZ2 vector spaces.The forgetful functorC ! D induces a functor' W HoC ! HoD. To see this,let A0

�, B0

�be filtered complexes, and letA� D '.A0

�/ andB� D '.B0

�/. A

quasi-isomorphismf WA0�!B0

�induces an isomorphism of the corresponding

spectral sequences, which implies thatf induces an isomorphismH�.A�/!

H�.B�/; in other wordsf WA�! B� is a quasi-isomorphism.Let C� W Schc.R/! HoD be the functor that assigns to every real algebraic

varietyX the complex of semialgebraic chainsC�.X /. ThenC� satisfies prop-erties (1) and (2) of Theorem 1.1. Acyclicity ofC� for an acyclic square (1-2)follows from the short exact sequence of chain complexes

0! C�. QY /! C�.Y /˚C�. QX /! C�.X /! 0:

The exactness of this sequence follows immediately from thedefinition of semi-algebraic chains. Similarly, additivity ofC� for a closed embeddingY ! X

follows from the short exact sequence of chain complexes

0! C�.Y /! C�.X /! C�.X nY /! 0:

Now consider the functorWC� WSchc.R/!HoC given by Theorem 1.1. Thefunctors'ıWC� andC� WSchc.R/!HoD are extensions ofC� WV.R/!HoD,so by [15] Theorem.2:2:2/op we have that'.WC�.X // is quasi-isomorphic toC�.X / for all X . ThusH�.WC�.X //DH�.X /, as desired. ˜


1C. The weight spectral sequence.If X is a real algebraic variety, theweightspectral sequenceof X , fEr ; dr g, r D 1; 2; : : : , is the spectral sequence of theweight complexWC�.X /. It is well-defined by Theorem 1.1, and it converges tothe homology ofX by Proposition 1.5. The associated filtration of the homologyof X is theweight filtration:

0DW�k�1Hk.X /�W�kHk.X /� � � � �W0Hk.X /DHk.X /;

whereHk.X / is the homology with closed supports (Borel–Moore homology)with coefficients inZ2. (We show thatW�k�1Hk.X /D 0 in Corollary 1.10.)The dual weight filtration on cohomology with compact supports is discussedin [25].

REMARK 1.6. We do not know the relation of the weight filtration of a realalgebraic varietyX to Deligne’s weight filtration [10] onH�.XCIQ/, the Borel–Moore homology with rational coefficients of the complex pointsXC . By anal-ogy with Deligne’s weight filtration, there should also be a weight filtrationon the homology ofX with classical compact supports and coefficients inZ2

(dual to cohomology with closed supports). We plan to study this filtration insubsequent work.

The weight spectral sequenceErp;q is a second quadrant spectral sequence. (We

will show in Corollary 1.10 that ifE1p;q¤0 then.p; q/ lies in the closed triangle

with vertices.0; 0/, .0; d/, .�d; 2d/, whered D dimX .) The reindexing

p0 D 2pC q; q0 D�p; r 0 D r C 1

gives a standard first quadrant spectral sequence, with

QE2p0;q0 DE1

�q0;p0C2q0 :

(If QE2p0;q0 ¤ 0 then.p0; q0/ lies in the closed triangle with vertices.0; 0/, .d; 0/,

.0; d/, whered D dimX .) Note that the total grading is preserved:p0C q0 D

pC q.The virtual Betti numbers [25] are the Euler characteristics of the rows ofQE2, that is,

ˇq.X /DX

p

.�1/p dimZ2QE2p;q : (1-4)

To prove this assertion we will show that the numbersˇq.X / defined by (1-4) areadditive and equal to the classical Betti numbers forX compact and nonsingular.

For eachq � 0 consider the chain complex defined by theq-th row of the QE1

term,

C�.X; q/D . QE1�;q;Qd1�;q/;


where Qd1p;q W

QE1p;q !

QE1p�1;q

. This chain complex is well-defined up to quasi-isomorphism, and its Euler characteristic isˇq.X /.

The additivity ofWC� implies that if Y is a closed subvariety ofX thenthe chain complexC�.X nY; q/ is quasi-isomorphic to the mapping cone of thechain mapC�.Y; q/! C�.X; q/, and hence there is a long exact sequence ofhomology groups

� � � ! QE2p;q.Y /!

QE2p;q.X /!

QE2p;q.X nY /! QE2

p�1;q.Y / � � � :

Therefore for eachq we have

ˇq.X /D ˇq.X nY /Cˇq.Y /:

This is the additivity property of the virtual Betti numbers.

REMARK 1.7. Navarro Aznar pointed out to us thatC�.X; q/ is actually well-defined up to chain homotopy equivalence. One merely applies[15], Theorem.2:2:2/op, to the functor that assigns to a nonsingular projective variety M thechain complex

Ck.M; q/D

�

Hq.M / if k D 0,0 if k ¤ 0,

in the category of bounded complexes ofZ2 vector spaces localized with re-spect to chain homotopy equivalences. This striking application of the theoremof Guillen and Navarro Aznar led to our proof of the existence of the weightcomplex.

We say the weight complex ispure if the reindexed weight spectral sequencehas QE2

p;q D 0 for p ¤ 0. In this case the numbersq.X / equal the classicalBetti numbers ofX .

PROPOSITION1.8. If X is a compact nonsingular variety, the weight complexWC �.X / is pure. In other words, if k ¤�p then

Hk

�

WpC�.X /

Wp�1C�.X /

�

D 0:

PROOF. For X projective and nonsingular, the filtered complexWC �.X / isquasi-isomorphic toC�.X /with the canonical filtration. The inclusionV.R/!Reg has the extension property in (2.1.10) of [15]; the proofis similar to that in(2.1.11) of the same reference. Therefore by Theorem.2:1:5/op [15], the functorF canC� WV.R/!HoC extends to a functor Reg!HoC that is additive for dis-joint unions and acyclic, and this extension is unique up to quasi-isomorphism.But F canC� W Reg! HoC is such an extension, sinceF canC� is additive fordisjoint unions in Reg and acyclic for acyclic squares in Reg. (Compare theproof of Theorem 1.1 and Remark 1.2.) ˜


If X is compact, we will show that the reindexed weight spectral sequenceQErp;q

is isomorphic to the spectral sequence of acubical hyperresolutionof X [15].(The definition of cubical hyperresolution given in Chapter5 of [34] is too weakfor our purposes; see Example 1.12 below.)

A cubical hyperresolution ofX is a special type of Cn -diagram with final

objectX and all other objects compact and nonsingular. RemovingX gives a˜n-diagram, which is the same thing as a4n-diagram,i.e.a diagram labeled bythe simplices contained in the standardn-simplex4n. (Subsets off0; 1; : : : ; ngof cardinalityi C 1 correspond toi-simplices.)

The spectral sequence of a cubical hyperresolution is the spectral sequenceof the filtered complex.C�; OF /, with Ck D

L

iCjDk Cj X .i/, whereX .i/ is thedisjoint union of the objects labeled byi-simplices of4n, and the filtration OFis by skeletons,

OFpCk DM

i�p

Ck�iX.i/

The resulting first quadrant spectral sequenceOErp;q converges to the homology

of X , and the associated filtration is the weight filtration defined by Totaro [37].Let @ D @0 C @00 be the boundary operator of the complexC�, where@0

i W

Cj X .i/ ! Cj X .i�1/ is the simplicial boundary operator, and@00

j W Cj X .i/ !

Cj�1X .i/ is .�1/i times the boundary operator on semialgebraic chains.

PROPOSITION1.9. If X is a compact variety, the weight spectral sequenceE ofX is isomorphic to the spectral sequenceOE of a cubical hyperresolution ofX :

Erp;q Š

OErC12pCq;�p

:

Thus OErp;q Š

QErp;q, the reindexed weight spectral sequence introduced above.

PROOF. The acyclicity property of the weight complex — condition (1) of The-orem 1.1 — implies thatWC � is acyclic for cubical hyperresolutions (see [15],proof of Theorem (2.1.5)). In other words, if the functorW C� is applied to acubical hyperresolution ofX , the resulting C

n -diagram inC is acyclic. Thissays thatWC �.X / is filtered quasi-isomorphic to the total filtered complex ofthe double complexWC i;j DWCj X .i/. Since the varietiesX .i/ are compactand nonsingular, this filtered complex is quasi-isomorphicto the total complexCk D

L

iCjDk Cj X .i/ with the canonical filtration,

F canp Ck D Ker@00

�p˚M

j>�p

Cj X .k�j/:

Thus the spectral sequence of this filtered complex is the weight spectral se-quenceEr

p;q.


We now compare the two increasing filtrationsF can and OF on the complexC�. The weight spectral sequenceE is associated to the filtrationF can, and thecubical hyperresolution spectral sequenceOE is associated to the filtrationOF . Weshow thatF canDDec. OF /, theDeligne shiftof OF ; for this notion see [8, (1.3.3)]or [34, A.49].

Let OF 0 be the filtration

OF 0

pCk D OZ1p;k�p D KerŒ@ W OFpCk ! Ck�1= OFp�1Ck�1�

and OE0 the associated spectral sequence. By definition of the Deligne shift,

OF 0

pCk D Dec OFp�kCk :

Now since@D @0C @00 it follows that

OF 0

pCk D F canp�kCk ;

andF canp�k

Ck D F can�q Ck , wherepC q D k. Thus we can identify the spectral

sequences. OE0/rC1

p;q DEr�q;pC2q for r � 1:

On the other hand, the inclusionOF 0pCk ! OFpCk induces an isomorphism of

spectral sequences. OE0/rp;q Š

OErp;q for r � 2: ˜

COROLLARY 1.10. Let X be a real algebraic variety of dimensiond , withweight spectral sequenceE and weight filtrationW. For all p; q; r , if Er

p;q ¤ 0

thenp � 0 and�2p � q � d �p. Thus for allk we haveW�k�1Hk.X /D 0.

PROOF. For X compact this follows from Proposition 1.9 and the fact thatOErp;q ¤ 0 impliesp � 0 and0� q � d �p. If U is a noncompact variety, letX

be a real algebraic compactification ofU , and letY D X nU . We can assumethat dimY < d . The corollary now follows from the additivity property of theweight complex (condition (2) of Theorem 1.1). ˜

EXAMPLE 1.11. If X is a compact divisor with normal crossings in a nonsin-gular variety, a cubical hyperresolution ofX is given by the decomposition ofX into irreducible components. (The corresponding simplicial diagram asso-ciates to ani-simplex the disjoint union of the intersections ofi C 1 distinctirreducible components ofX .) The spectral sequence of such a cubical hyper-resolution is the Mayer–Vietoris (orCech) spectral sequence associated to thedecomposition. Example 3.3 of [25] is an algebraic surfaceX in affine 3-spacesuch thatX is the union of three compact nonsingular surfaces with normalcrossings and the weight spectral sequence ofX does not collapse:QE2 ¤ QE1.The varietyU D R

3 n X is an example of a nonsingular noncompact variety


with noncollapsing weight spectral sequence. (The additivity property (2) ofTheorem 1.1 can be used to compute the spectral sequence ofU .)

EXAMPLE 1.12. For a compact complex variety the Deligne weight filtrationcan be computed from the skeletal filtration of a simplicial smooth resolution ofcohomological descent(see [9, (5.3)] or [34, (5.1.3)]). In particular, a rationalhomology class has maximal weight if and only if is in the image of thehomology of the zero-skeleton of the resolution.

The following example shows that for real varieties the cohomological de-scent condition on a resolution is too weak to recover the weight filtration.

We construct a simplicial smooth varietyX�!X of cohomological descentsuch thatX is compact and the weight filtration ofX does not correspond tothe skeletal filtration ofX�. Let X D X0 D S1, the unit circle in the complexplane, and letf W X0 ! X be the double coverf .z/ D z2. Let X� be theGabrielov–Vorobjov–Zell resolution associated to the mapf [13] . Thus

Xn DX0 �X X0 �X � � � .nC 1/ � � � �X X0;

a compact smooth variety of dimension 1. This resolution is of cohomologicaldescent since the fibers of the geometric realizationjX�j ! X are contractible(see [13] or [34, (5.1.3)]).

Let˛2H1.X / be the nonzero element (Z2 coefficients). Now 2W�1H.X /

sinceX is compact and nonsingular. Therefore, for every cubical hyperresolu-tion of X , ˛ lies in the image of the homology of the zero-skeleton (i.e., thefiltration of ˛ with respect to the spectral sequenceOE is 0). But the filtration of˛ with respect to the skeletons of the resolutionX�!X is greater than 0 since˛ … ImŒf� W H1.X0/! H1.X /�. In fact ˛ has filtration 1 with respect to theskeletons of this resolution.

2. A geometric filtration

We define a functorGC� W Schc.R/! C

that assigns to each real algebraic varietyX the complexC�.X / of semialge-braic chains ofX (with coefficients inZ2 and closed supports), together with afiltration

0D G�k�1Ck.X /� G�kCk.X /� G�kC1Ck.X /

� � � � � G0Ck.X /D Ck.X /: (2-1)

We prove in Theorem 2.8 that the functorGC� realizes the weight complexfunctorWC� W Schc.R/! HoC given by Theorem 1.1. Thus the filtrationG�

of chains gives the weight filtration of homology.


2A. Definition of the filtration G�. The filtration will first be defined for com-pact varieties. Recall thatX denotes the set of real points of the real algebraicvarietyX .

THEOREM 2.1. There exists a unique filtration(2-1) on semialgebraicZ2-chains of compact real algebraic varieties with the following properties. LetX be a compact real algebraic variety and letc 2 Ck.X /. Then

(1) If Y �X is an algebraic subvariety such thatSuppc � Y , then

c 2 GpCk.X / ” c 2 GpCk.Y /:

(2) Let dimX D k and let� W QX !X be a resolution ofX such that there is anormal crossing divisorD � QX with Supp@.��1c/�D. Then forp � �k,

c 2 GpCk.X / ” @.��1c/ 2 GpCk�1.D/:

We call a resolution� W QX ! X adaptedto c 2 Ck.X / if it satisfies condition(2) above. For the definition of the support Suppc and the pullback��1c seethe Appendix.

PROOF. We proceed by induction onk. If k D 0 then 0 D G�1C0.X / �

G0C0.X / D C0.X /. In the rest of this subsection we assume the existenceand uniqueness of the filtration for chains of dimension< k, and we prove thestatement for chains of dimensionk.

LEMMA 2.2. Let X DSs

iD1 Xi whereXi are subvarieties ofX . Then form< k,

c 2 GpCm.X / ” cjXi2 GpCm.Xi/ for all i:

PROOF. By (1) we may assume that dimX Dm and then that allXi are distinctof dimensionm. Thus an adapted resolution ofX is a collection of adaptedresolutions of each component ofX . ˜

See the Appendix for the definition of the restrictioncjXi.

PROPOSITION2.3. The filtrationGp given by Theorem2.1 is functorial; thatis, for a regular morphismf W X ! Y of compact real algebraic varieties,f�.GpCm.X //� GpCm.Y /, for m< k.

PROOF. We prove that if the filtration satisfies the statement of Theorem 2.1 forchains of dimension< k and is functorial on chains of dimension< k � 1 thenit is functorial on chains of dimensionk � 1.

Let c 2Ck�1.X /, and letf WX ! Y be a regular morphism of compact realalgebraic varieties. By (1) of Theorem 2.1 we may assume dimX D dimY D


k � 1 and by Lemma 2.2 thatX andY are irreducible. We may assume thatf

is dominant; otherwisef�c D 0. Then there exists a commutative diagram

QXQf

��! QY

�X

?

?

y

?

?

y

�Y

Xf

��! Y

where�X is a resolution ofX adapted toc and�Y a resolution ofY adaptedto f�c. Then

c 2 Gp.X / () @.��1X c/ 2 Gp. QX / ) Qf� @.�

�1X c/ 2 Gp. QY /;

Qf� @.��1X c/D @ Qf�.�

�1X c/D @.��1

Y f�c/;

@.��1Y f�c/ 2 Gp. QY / () f�c 2 Gp.Y /;

where the implication in the first line follows from the inductive assumption.

COROLLARY 2.4. The boundary operator@ preserves the filtrationGp:

@GpCm.X //� GpCm�1.X / for m< k:

PROOF. Let� W QX !X be a resolution ofX adapted toc. Let QcD ��1c. Thenc D �� Qc and

c 2 Gp () @ Qc 2 Gp ) @c D @�� Qc D ��@ Qc 2 Gp: ˜

Let c 2 Ck.X /, dimX D k. In order to show that condition (2) of Theorem 2.1is independent of the choice ofQ� we need the following lemma.

LEMMA 2.5. Let X be a nonsingular compact real algebraic variety of dimen-sion k and letD � X be a normal crossing divisor. Let c 2 Ck.X / satisfySupp@c�D. Let� W QX!X be the blowup of a nonsingular subvarietyC �X

that has normal crossings withD. Then

@c 2 GpCk�1.X / ” @.��1.c// 2 GpCk�1. QX /:

PROOF. Let QD D ��1.D/. Then QD D E [S

QDi , whereE D ��1.C / is theexceptional divisor andQDi denotes the strict transform ofDi . By Lemma 2.2,

@c 2 GpCk�1.X / ” @cjDi2 GpCk�1.Di/ for all i:

Let@icD@cjDi. The restriction�iD�j QDi

W QDi!Di is the blowup with smoothcenterC \Di . Hence, by the inductive assumption,

@.@ic/ 2 GpCk�2.Di/ ” @��1i .@ic/D @

�

@.��1.c//j QDi

�

2 GpCk�2. QDi/


By the inductive assumption of Theorem 2.1,

@.@ic/ 2 GpCk�2.Di/ ” @ic 2 GpCk�1.Di/;

and we have similar properties for@.��1.c//j QDiand@.��1.c//jE.

Thus, to complete the proof it suffices to show that if@�

@.��1.c//j QDi

�

lies inGpCk�2. QDi/ for all i , then@

�

@.��1.c//jE�

2 GpCk�2.E/. This follows from

0D @.@��1.c//D @�

P

i

@.��1.c//j QDiC @.��1.c//jE

�

: ˜

Let �i W Xi ! X; i D 1; 2; be two resolutions ofX adapted toc. Then thereexists� W QX1 ! X1, the composition of finitely many blowups with smoothcenters that have normal crossings with the strict transforms of all exceptionaldivisors, such that�1 ı � factors throughX2,

QX1�

��! X1

�

?

?

y

?

?

y

�1

X2

�2

��! X

By Lemma 2.5,

@.��11 .c// 2 GpCk�1.X1/ () @.��1.��1

1 .c/// 2 GpCk�1. QX1/:

On the other hand,

��@.��1.��1

1 .c///D ��@.��1.��1

2 .c///D @.��12 .c//;

and consequently by Proposition 2.3 we have

@.��11 .c// 2 GpCk�1.X1/ ÷ @.��1

2 .c// 2 GpCk�1.X2/:

By symmetry,@.��12.c//2Gp.X / implies@.��1

1.c//2Gp.X /. This completes

the proof of Theorem 2.1. ˜

2B. Properties of the filtration G�. Let U be a (not necessarily compact) realalgebraic variety and letX be a real algebraic compactification ofU . We extendthe filtrationGp to U as follows. Ifc 2C�.U /, let Nc 2C�.X / be its closure. Wedefine

c 2 GpCk.U / () Nc 2 GpCk.X /:

See the Appendix for the definition of the closure of a chain.

PROPOSITION2.6.GpCk.U / is well-defined; that is, for two compactificationsX1 andX2 of U , we have

c1 2 GpCk.X1/ () c2 2 GpCk.X2/;

whereci denotes the closure ofc in Xi , i D 1; 2.


PROOF. We may assume thatk D dimU . By a standard argument, any twocompactifications can be dominated by a third one. Indeed, denote the inclusionsby ii WU ŒXi . Then the Zariski closureX of the image of.i1; i2/ in X1�X2

is a compactification ofU .Thus we may assume that there is a morphismf WX2!X1 that is the identity

onU . Then, by functoriality,c2 2 GpCk.X2/ impliesc1D f�.c2/2 GpCk.X1/.By the Chow–Hironaka lemma there is a resolution�1 W QX1! X1, adapted toc1, that factors throughf : �1 D f ı g. Thenc1 2 GpCk.X1/ is equivalent to��1

1.c1/ 2 GpCk. QX1/; but this implies thatc2 D g�.�

�11.c1// 2 GpCk.X2/, as

needed. ˜

THEOREM 2.7. The filtrationG� defines a functorGC� W Schc.R/! C with thefollowing properties:

(1) For an acyclic square(1-2) the following sequences are exact:

0! GpCk. QY /! GpCk.Y /˚GpCk. QX /! GpCk.X /! 0;

0!GpCk. QY /

Gp�1Ck. QY /!

GpCk.Y /

Gp�1Ck.Y /˚

GpCk. QX /

Gp�1Ck. QX /!

GpCk.X /

Gp�1Ck.X /! 0:

(2) For a closed inclusionY � X , with U D X n Y , the following sequencesare exact:

0! GpCk.Y /! GpCk.X /! GpCk.U /! 0;

0!GpCk.Y /

Gp�1Ck.Y /!

GpCk.X /

Gp�1Ck.X /!

GpCk.U /

Gp�1Ck.U /! 0:

PROOF. The exactness of the first sequence of (2) follows directly from thedefinitions (moreover, this sequence splits viac ‘ Nc). The exactness of thesecond sequence of (2) now follows by a diagram chase. Similarly, the exactnessof the first sequence of (1) follows from the definitions, and the exactness of thesecond sequence of (1) is proved by a diagram chase. ˜

For any varietyX , the filtrationG� is contained in the canonical filtration,

GpCk.X /� F canp Ck.X /; (2-2)

[email protected]�kCk.X //D 0. Thus on the category of nonsingular projective vari-eties we have a morphism of functors

� W GC�! F canC�:

THEOREM 2.8. For every nonsingular projective real algebraic varietyM ,

�.M / W GC�.M /! F canC�.M /


is a filtered quasi-isomorphism. Hence, for every real algebraic varietyX thelocalization of� induces a quasi-isomorphism� 0.X / W GC�.X /!WC�.X /.

Theorem 2.8 follows from Corollary 3.11 and Corollary 3.12,which will beshown in the next section.

3. The Nash constructible filtration

In this section we introduce theNash constructible filtration

0DN�k�1Ck.X /�N�kCk.X /�N�kC1Ck.X /

� � � � �N0Ck.X /D Ck.X / (3-1)

on the semialgebraic chain complexC�.X / of a real algebraic varietyX . Weshow that this filtration induces a functor

NC� W Schc.R/! C

that realizes the weight complex functorWC� W Schc.R/ ! HoC. In orderto prove this assertion in Theorem 3.11, we have to extendNC� to a widercategory of sets and morphisms. The objects of this categoryare certain semi-algebraic subsets of the set of real points of a real algebraic variety, and theyinclude in particular all connected components of real algebraic subsets ofRPn.The morphisms are certain proper continuous semialgebraicmaps between thesesets. This extension is crucial for the proof. As a corollarywe show that forreal algebraic varieties the Nash constructible filtrationN� coincides with thegeometric filtrationG� of Section 2A. In this way we complete the proof ofTheorem 2.8.

For real algebraic varieties, the Nash constructible filtration was first definedin an unpublished paper of H. Pennaneac’h [32], by analogy with the alge-braically constructible filtration [31; 33]. Theorem 3.11 implies, in particular,that the Nash constructible filtration of a compact variety is the same as the fil-tration given by a cubical hyperresolution; this answers affirmatively a questionof Pennaneac’h [32, (2.9)].

3A. Nash constructible functions onRPn and arc-symmetric sets. In realalgebraic geometry it is common to work with real algebraic subsets of theaffine spaceRn � RPn instead of schemes overR, and with (entire) regularrational mappings as morphisms; see for instance [3] or [5].SinceRPn can beembedded inRN by a biregular rational map ([3], [5] (3.4.4)), this category alsocontains algebraic subsets ofRPn.

A Nash constructible functionon RPn is a function' W RPn! Z such thatthere exist a finite family of regular rational mappingsfi WZi!RPn defined on


projective real algebraic setsZi , connected componentsZ0

i of Zi , and integersmi , such that for allx 2 RPn,

'.x/DX

i

mi�.f�1

i .x/\Z0

i/; (3-2)

where� is the Euler characteristic. Nash constructible functionswere intro-duced in [24]. Nash constructible functions onRPn form a ring.

EXAMPLE 3.1.

(1) If Y � RPn is Zariski constructible (a finite set-theoretic combination ofalgebraic subsets), then its characteristic function1Y is Nash constructible.

(2) A subsetS � RPn is calledarc-symmetricif every real analytic arc W.a; b/! RPn either meetsS at isolated points or is entirely included inS .Arc-symmetric sets were first studied by K. Kurdyka in [19]. As shown in[24], a semialgebraic setS �RPn is arc-symmetric if and only if it is closedin RPn and 1S is Nash constructible. By the existence of arc-symmetricclosure [19; 21], for a setS �RPn the function1S is Nash constructible andonly if S is a finite set-theoretic combination of semialgebraic arc-symmetricsubsets ofRPn. If 1S is Nash constructible we say thatS is anAS set.

(3) Any connected component of a compact algebraic subset ofRPn is arc-symmetric. So is any compact real analytic and semialgebraic subset ofRPn.

(4) Every Nash constructible function onRPn is in particularconstructible(constant on strata of a finite semialgebraic stratificationof RPn). Not allconstructible functions are Nash constructible. By [24], every constructiblefunction' W RPn! 2nZ is Nash constructible.

Nash constructible functions form the smallest family of constructible functionsthat contains characteristic functions of connected components of compact realalgebraic sets, and that is stable under the natural operations inherited fromsheaf theory: pullback by regular rational morphisms, pushforward by properregular rational morphisms, restriction to Zariski open sets, and duality; see[24]. In terms of thepushforward(fiberwise integration with respect to theEuler characteristic) the formula (3-2) can be expressed as' D

P

i mifi �1Z 0i.

Duality is closely related to thelink operator, an important tool for studyingthe topological properties of real algebraic sets. For moreon Nash constructiblefunction see [7] and [21].

If S �RPn is anAS set (i.e. 1S is Nash constructible), we say that a functiononS is Nash constructibleif it is the restriction of a Nash constructible functionon RPn. In particular, this defines Nash constructible functions on affine realalgebraic sets. (In the non-compact case this definition is more restrictive thanthat of [24].)


3B. Nash constructible functions on real algebraic varieties. Let X be areal algebraic variety and letX denote the set of real points onX . We call afunction' W X ! Z Nash constructibleif its restriction to every affine chart isNash constructible. The following lemma shows that this extends our definitionof Nash constructible functions on affine real algebraic sets.

LEMMA 3.2. If X1 and X2 are projective compactifications of the affine realalgebraic varietyU , then' W U ! Z is the restriction of a Nash constructiblefunction onX 1 if and only if' is the restriction of a Nash constructible functiononX 2.

PROOF. We may suppose that there is a regular projective morphismf WX1!

X2 that is an isomorphism onU ; cf. the proof of Proposition 2.6. Then the state-ment follows from the following two properties of Nash constructible functions.If '2 WX 2!Z is Nash constructible, so is its pullbackf �'2D'2ıf WX 1!Z.If '1 WX 1! Z is Nash constructible, so is its pushforwardf�'1 WX 2! Z. ˜

PROPOSITION3.3. LetX be a real algebraic variety and letY �X be a closedsubvariety. Let U D X n Y . Then' W X ! Z is Nash constructible if and onlyif the restrictions of' to Y andU are Nash constructible.

PROOF. It suffices to check the assertion forX affine; this case is easy. ˜

THEOREM 3.4. Let X be a complete real algebraic variety. The function' W X ! Z is Nash constructible if and only if there exist a finite family ofregular morphismsfi W Zi ! X defined on complete real algebraic varietiesZi , connected componentsZ0

i of Zi , and integersmi , such that for allx 2X ,

' DX

i

mi fi �1Z 0i: (3-3)

PROOF. If X is complete but not projective, thenX can be dominated by abirational regular morphism� W QX ! X , with QX projective (Chow’s Lemma).Let Y � X , dimY < dimX , be a closed subvariety such that� induces anisomorphism QX n��1.Y /!X nY . Then, by Proposition 3.3,' WX!Z is Nashconstructible if and only if��' and' restricted toY are Nash constructible.

Let Z be a complete real algebraic variety and letf W Z ! X be a regularmorphism. LetZ0 be a connected component ofZ. We show that' D f�1Z 0

is Nash constructible. This is obvious if bothX andZ are projective. If theyare not, we may dominate bothX andZ by projective varieties, using Chow’sLemma, and reduce to the projective case by induction on dimension.

Let' WX!Z be Nash constructible. Suppose first thatX is projective. ThenX � RPn is a real algebraic set. LetA � RPm be a real algebraic set and letf WA! X be a regular rational morphismf D g=h, whereh does not vanishon A, cf. [3]. Then the graph off is an algebraic subset� � RPn � RPm


and the set of real points of a projective real varietyZ. Let A0 be a connectedcomponent ofA, and� 0 the graph off restricted toA0. Thenf�1A0 D ��1� 0 ,where� denotes the projection on the second factor.

If X is complete but not projective, we again dominate it by a birationalregular morphism� W QX ! X , with QX projective. Let' W X ! Z be Nashconstructible. ThenQ'D ' ı� W QX !Z is Nash constructible. Thus, by the caseconsidered above, there are regular morphismsQfi W QZi ! QX , and connectedcomponentsQZ0

i such that

Q'.x/DX

i

miQfi �1 QZ 0

i

:

Then�� Q' DP

i mi Q��fi �1 QZ 0i

and differs from' only on the set of real points

of a variety of dimension smaller than dimX . We complete the argument byinduction on dimension. ˜

If X is a real algebraic variety, we again say thatS � X is anAS set if 1S isNash constructible, and' W S ! Z is Nash constructibleif the extension of'to X by zero is a Nash constructible function onX .

COROLLARY 3.5. Let X;Y be complete real algebraic varieties and letS beanAS subset ofX , andT anAS subset ofY . Let' W S ! Z and W T ! Z

be Nash constructible. Let f W S ! T be a map withAS graph� � X � Y

and let�X WX �Y !X and�Y WX �Y ! Y denote the standard projections.Then

f�.'/D .�Y /�.1� ��

X '/ (3-4)

andf �. /D .�X /�.1� ��

�

Y / (3-5)

are Nash constructible.

3C. Definition of the Nash constructible filtration. Denote byXAS the cat-egory of locally compactAS subsets of real algebraic varieties as objects andcontinuous proper maps withAS graphs as morphisms.

Let T 2 XAS . We say that' W T ! Z is generically Nash constructible onT in dimensionk if ' coincides with a Nash constructible function everywhereon T except on a semialgebraic subset ofT of dimension< k. We say that'is generically Nash constructible onT if ' is Nash constructible in dimensiond D dimT .

Let c 2 Ck.T /, and let�k � p � 0. We say thatc is p-Nash constructible,and writec 2 NpCk.T /, if there exists'c;p W T ! 2kCpZ, generically Nashconstructible in dimensionk, such that

c D fx 2 T I 'c;p.x/ … 2kCpC1Zg up to a set of dimension< k. (3-6)


up to a set of dimension less thank. The choice of'c;p is not unique. LetZ denote the Zariski closure of Suppc. By multiplying 'c;p by 1Z , we mayalways assume that Supp'�Z and hence, in particular, that dim Supp'c;p�k.

We say thatc 2Ck.T / is pureif c 2N�kCk.T /. By Theorem 3.9 of [21] andthe existence of arc-symmetric closure [19; 21],c 2Ck.T / is pure if and only ifSuppc coincides with anAS set (up to a set of dimension smaller thank). ForT compact this means thatc is pure if and only ifc can be represented by an arc-symmetric set. By [24], if dimT Dk then every semialgebraically constructiblefunction' W T ! 2kZ is Nash constructible. HenceN0Ck.T /D Ck.T /.

The boundary operator preserves the Nash constructible filtration:

@NpCk.T /�NpCk�1.T /:

Indeed, ifc 2 Ck.T / is given by (3-6) and dim Supp'c;p � k, then

@c D fx 2Z I '@c;p.x/ … 2kCpZg; (3-7)

where'@c;p equals12�'c;p for k odd and1

2˝'c;p for k even [24]. A geometric

interpretation of this formula is as follows; see [7]. LetZ be the Zariski closureof Suppc, so dimZ D k if c ¤ 0. Let W be an algebraic subset ofZ such thatdimW < k and'c;p is locally constant onZ nW . At a generic pointx of W ,we define@W 'c;p.x/ as the average of the values of'c;p on the local connectedcomponents ofZnW atx. It can be shown that@W 'c;p.x/ is generically Nashconstructible in dimensionk � 1. (For k odd it equals.1

2�'c;p/jW and fork

even it equals.12˝'c;p/jW ; see [24].)

We say that a square inXAS

QS ��! QT?

?

y

?

?

y

�

Si

��! T

(3-8)

is acyclic if i is a closed inclusion,QS D ��1.Y / and the restriction of� is ahomeomorphismQT n QS ! T nS .

THEOREM 3.6. The functorNC� W XAS ! C, defined on the categoryXAS oflocally compactAS sets and continuous proper maps withAS graphs, satisfies:

(1) For an acyclic square(3-8) the sequences

0!NpCk. QS/!NpCk.S/˚NpCk. QT /!NpCk.T /! 0;

0!NpCk. QS/

Np�1Ck. QS/!

NpCk.S/

Np�1Ck.S/˚

NpCk. QT /

Np�1Ck. QT /!

NpCk.T /

Np�1Ck.T /! 0;

are exact.


(2) For a closed inclusionS � T , the restriction toU D T n S induces amorphism of filtered complexesNC�.T /!NC�.U /, and the sequences

0!NpCk.S/!NpCk.T /!NpCk.U /! 0;

0!NpCk.S/

Np�1Ck.S/!

NpCk.T /

Np�1Ck.T /!

NpCk.U /

Np�1Ck.U /! 0;

are exact.

PROOF. We first show thatNC� is a functor; that is, for a proper morphismf W T ! S , f�NpCk.T / �NpCk.S/. Let c 2NpCk.T / and let' D 'c;p bea Nash constructible function onT satisfying (3-6) (up to a set of dimension< k). Then

f�c D fy 2 S I f�. /.y/ … 2kCpC1ZgI

that is,'f�c;p D f�'c;p.For a closed inclusionS � T , the restriction toU D T n S of a Nash con-

structible function onT is Nash constructible. Therefore the restriction definesa morphismNC�.T /! NC�.U /. The exactness of the first sequence of (2)can be verified easily by direct computation. We note, moreover, that for fixedk the morphism

N�Ck.T /!N�Ck.U /

splits (the splitting does not commute with the boundary), by assigning toc 2NpCk.U / its closureNc 2 Ck.T /. Let ' W T ! 2kCpZ be a Nash constructiblefunction such that'jT nSD'c;p. ThenNcDfx2T I .1T�1S /'.x/…2kCpC1Zg

up to a set of dimension< k.The exactness of the second sequence of (2) and the sequencesof (1) now

follow by standard arguments. (See the proof of Theorem 2.7.) ˜

3D. The Nash constructible filtration for Nash manifolds. A Nash func-tion on an open semialgebraic subsetU of R

N is a real analytic semialgebraicfunction. Nash morphisms and Nash manifolds play an important role in realalgebraic geometry. In particular a connected component ofcompact nonsingu-lar real algebraic subset ofR

n is a Nash submanifold ofRN in the sense of [5](2.9.9). SinceRPn can be embedded inRN by a rational diffeomorphism ([3],[5] (3.4.2)) the connected components of nonsingular projective real algebraicvarieties can be considered as Nash submanifolds of affine space. By the NashTheorem [5, 14.1.8], every compactC 1 manifold isC 1-diffeomorphic to aNash submanifold of an affine space, and moreover such a modelis uniqueup to Nash diffeomorphism [5, Corollary 8.9.7]. In what follows by aNashmanifoldwe mean a compact Nash submanifold of an affine space.


Compact Nash manifolds and the graphs of Nash morphisms on them areAS

sets. IfN is a Nash manifold, the Nash constructible filtration is contained inthe canonical filtration,

NpCk.N /� F canp Ck.N /; (3-9)

[email protected]�kCk.N //D 0. Thus on the category of Nash manifolds and Nashmaps have a morphism of functors

� WNC�! F canC�:

THEOREM 3.7. For every Nash manifoldN ,

�.N / WNC�.N /! F canC�.N /

is a filtered quasi-isomorphism.

PROOF. We show that for allp andk, �.N / induces an isomorphism

�� WHk.NpC�.N //ŠHk.Fcanp C�.N //: (3-10)

Then, by the long exact homology sequences of.NpC�.N /;Np�1C�.N // and.F can

p C�.N /; F canp�1

C�.N //,

�� WHk

�

NpC�.N /

Np�1C�.N /

�

!Hk

�

F canp C�.N /

F canp�1

C�.N /

�

is an isomorphism, which shows the claim of the theorem.We proceed by induction on the dimension ofN . We call a Nash morphism

� W QN!N aNash multi-blowupif � is a composition of blowups along nowheredense Nash submanifolds.

PROPOSITION 3.8. Let N;N 0 be compact connected Nash manifolds of thesame dimension. Then there exist multi-blowups� W QN !N , � W QN 0!N 0 suchthat QN and QN 0 are Nash diffeomorphic.

PROOF. By a theorem of Mikhalkin (see [26] and Proposition 2.6 in [27]), anytwo connected closedC 1 manifolds of the same dimension can be connectedby a sequence ofC 1 blowups and and then blowdowns with smooth centers.We show that thisC 1 statement implies an analogous statement in the Nashcategory.

Let M be a closedC 1 manifold. By the Nash–Tognoli Theorem there isa nonsingular real algebraic setX , a fortiori a Nash manifold, that isC 1-diffeomorphic toM . Moreover, by approximation by Nash mappings, any twoNash models ofM are Nash diffeomorphic; see Corollary 8.9.7 in [5]. Thus inorder to show Proposition 3.8 we need only the following lemma.


LEMMA 3.9. Let C �M be aC 1 submanifold of a closedC 1 manifoldM .Suppose thatM is C 1-diffeomorphic to a Nash manifoldN . Then there existsa Nash submanifoldD � N such that the blowupsBl.M;C / of M along C

andBl.N;D/ of N alongD are C 1-diffeomorphic.

Proof. By the relative version of Nash–Tognoli Theorem proved by Akbulutand King, as well as Benedetti and Tognoli (see for instance Remark 14.1.15in [5]), there is a nonsingular real algebraic setX and aC 1 diffeomorphism' WM!X such thatY D'.C / is a nonsingular algebraic set. Then the blowupsBl.M;C / of M alongC andBl.X;Y / of X alongY areC 1-diffeomorphic.Moreover, sinceX andN areC 1-diffeomorphic, they are Nash diffeomorphicby a Nash diffeomorphism W X !N . ThenBl.X;Y / andBl.N; .Y // areNash diffeomorphic. This proves the lemma and the proposition. ˜

LEMMA 3.10.LetN be a compact connected Nash manifold and let� W QN!N

denote the blowup ofN along a nowhere dense Nash submanifoldY . Then�.N / is a quasi-isomorphism if and only if�. QN / is a quasi-isomorphism.

PROOF. Let QY D ��1.Y / denote the exceptional divisor of�. For eachp

consider the diagram

�! HkC1.NpC�.N // �! Hk.NpC�. QY // �! Hk.NpC�.Y //˚Hk.NpC�. QN // �!?

?

?

y

?

?

?

y

?

?

?

y

�! HkC1.Fcanp C�.N // �! Hk.F

canp C�. QY // �! Hk.F

canp C�.Y //˚Hk.F

canC�. QN // �!

The top row is exact by Theorem 3.6. For all manifoldsN and for allp andk,we have

Hk.Fcanp C�.N //D

�

Hk.N / if k � �p,0 if k < �p,

so the short exact sequences (1-3) give that the bottom row isexact. The lemmanow follows from the inductive assumption and the Five Lemma. ˜

Consequently it suffices to show that�.N / is a quasi-isomorphism for a singleconnected Nash manifold of each dimensionn. We check this assertion for thestandard sphereSn by showing that

Hk.NpC�.Sn//D

n

Hk.Sn/ if k D 0 or n andp ��k,

0 otherwise.

Let c 2 NpCk.Sn/, k < n, be a cycle described as in (3-6) by the Nash

constructible function'c;p W Z ! 2kCpZ, whereZ is the Zariski closure ofSuppc. Thenc can be contracted to a point. More precisely, choosep 2SnnZ.ThenSn n fpg andR

n are isomorphic. Define a Nash constructible function


˚ WZ �R! 2kCpC1Z by the formula

˚.x; t/Dn

2'c;p.x/ if t 2 Œ0; 1�,0 otherwise.

Then

c � Œ0; 1�D f.x; t/ 2Z �R I ˚.x; t/ … 2kCpC2ZgI

soc� Œ0; 1� 2NpCkC1.Z�R/. The morphismf WZ�R!Rn, f .x; t/D tx,

is proper and fork > 0

@f�.c � Œ0; 1�/D f�.@c � Œ0; 1�/D c;

which shows thatc is a boundary inNpC�.Sn/. If k D 0 then@f�.c� Œ0; 1�/D

c � .degc/Œ0�.If c 2NpCn.S

n/ is a cycle, thenc is a cycle inCn.Sn/; that is, eitherc D 0

or c D ŒSn�. This completes the proof of Theorem 3.7. ˜

3E. Consequences for the weight filtration.

COROLLARY 3.11. For every real algebraic varietyX the localization of�induces a quasi-isomorphism� 0.X / WNC�.X /!WC�.X /.

PROOF. Theorem 3.6 yields that the functorNC� W Schc.R/! HoC satisfiesproperties (1) and (2) of Theorem 1.1. Hence Theorem 3.7 and Theorem 1.4give the desired result. ˜

COROLLARY 3.12. Let X be a real algebraic variety. Then for allp and k,NpCk.X /D GpCk.X /.

PROOF. We show that the Nash constructible filtration satisfies properties (1)and (2) of Theorem 2.1. This is obvious for property (1). We show property (2).Let Qc D ��1.c/. First we note that

c 2NpCk.X / () Qc 2NpCk. QX /:

Indeed, (() follows from functoriality, sincec D ��. Qc/. If c is given by (3-1)then��.'c;p/ is Nash constructible and describesQc. Thus it suffices to show

Qc 2NpCk. QX / () @ Qc 2NpCk�1. QX /

for p � �k, with the implication () ) being obvious. Ifp D �k then eachcycle is arc-symmetric. (Such a cycle is a union of connectedcomponents ofQX , since QX is nonsingular and compact.) Forp > �k suppose, contrary to our

claim, that

Qc 2NpCk. QX / nNp�1Ck. QX / and @ Qc 2Np�1Ck�1. QX /:


By Corollary 3.11 and Proposition 1.8

Hk

�

NpC�. QX /

Np�1C�. QX /

�

D 0;

and Qc has to be a relative boundary. But dimQX D k andCkC1. QX / D 0. Thiscompletes the proof. ˜

4. Applications to real algebraic and analytic geometry

Algebraic subsets of affine space, or more generallyZ-open orZ-closedaffine or projective sets in the sense of Akbulut and King [3],areAS sets. Soare the graphs of regular rational mappings. Therefore Theorems 3.6 and 3.7give the following result.

THEOREM4.1. The Nash constructible filtration of closed semialgebraic chainsdefines a functor from the category of affine real algebraic sets and proper regu-lar rational mappings to the category of bounded chain complexes ofZ2 vectorspaces with increasing bounded filtration.

This functor is additive and acyclic; that is, it satisfies properties(1) and(2) of Theorem3.6; and it induces the weight spectral sequence and the weightfiltration on Borel–Moore homology with coefficients inZ2.

For compact nonsingular algebraic sets, the reindexed weight spectral se-quence is pure: QE2

p;q D 0 for p > 0.

For the last claim of the theorem we note that every compact affine real algebraicset that is nonsingular in the sense of [3] and [5] admits a compact nonsingularcomplexification. Thus the claim follows from Theorem 3.7.

The purity of QE2 implies the purity of QE1: QE1p;qD0 for p>0. Consequently

every nontrivial homology class of a nonsingular compact affine or projectivereal algebraic variety can be represented by a semialgebraic arc-symmetric set,a result proved directly in [18] and [21].

REMARK 4.2. Theorem 3.6 and Theorem 3.7 can be used in more generalcontexts. A compact real analytic semialgebraic subset of areal algebraic va-riety is anAS set. A compact semialgebraic set that is the graph of a realanalytic map, or more generally the graph of an arc-analyticmapping (cf. [21]),is arc-symmetric. In Section 3E we have already used that compact affine Nashmanifolds and graphs of Nash morphisms defined on compact Nash manifoldsare arc-symmetric.

The weight filtration of homology is an isomorphism invariant but not a home-omorphism invariant; this is discussed in [25] for the dual weight filtration ofcohomology.


PROPOSITION4.3. LetX andY be locally compactAS sets, and letf WX!Y

be a homeomorphism withAS graph. Thenf� W NC�.X /! NC�.Y / is anisomorphism of filtered complexes.

Consequently, f� induces an isomorphism of the weight spectral sequencesof X andY and of the weight filtrations ofH�.X / andH�.Y /. Thus the virtualBetti numbers(1-4)of X andY are equal.

PROOF. The first claim follows from the fact thatNC� WXAS! C is a functor;see the proof of Theorem 3.6. The rest of the proposition thenfollows fromTheorem 3.6 and Theorem 3.7. ˜

REMARK 4.4. Proposition 4.3 applies, for instance, to regular homeomorphismssuch asf W R! R, f .x/D x3. The construction of the virtual Betti numbersof [25] was extended toAS sets by G. Fichou in [11], where their invarianceby Nash diffeomorphism was shown. The arguments of [25] and [11] use theweak factorization theorem of [1].

4A. The virtual Poincare polynomial. Let X be a locally compactAS set.The virtual Betti numbers give rise to thevirtual Poincare polynomial

ˇ.X /DX

i

ˇi.X / t i : (4-1)

For real algebraic varieties the virtual Poincare polynomial was first introducedin [25]. ForAS sets, not necessarily locally compact, it was defined in [11]. Itsatisfies the following properties [25; 11]:

(i) Additivity: For finite disjoint unionX DF

Xi , we have .X /DP

ˇ.Xi/.(ii) Multiplicativity: ˇ.X �Y /D ˇ.X / �ˇ.Y /.(iii) Degree:For X ¤?, degˇ.X /D dimX and the leading coefficient.X /

is strictly positive.

(If X is not locally compact we can decompose it into a finite disjoint union oflocally compactAS setsX D

F

Xi and define .X /DP

ˇ.Xi/.)We say that a functionX!e.X / defined on real algebraic sets is aninvariant

if it an isomorphism invariant, that ise.X /D e.Y / if X andY are isomorphic(by a biregular rational mapping). We say thate is additive if e takes valuesin an abelian group ande.X n Y / D e.X /� e.Y / for all Y � X . We saye ismultiplicative if e takes values in a ring ande.X �Y /D e.X /e.Y / for all X;Y .The following theorem states that the virtual Betti polynomial is a universaladditive, or additive and multiplicative, invariant defined on real algebraic sets(or real points of real algebraic varieties in general), among those invariants thatdo not distinguish Nash diffeomorphic compact nonsingularreal algebraic sets.


THEOREM 4.5. Let e be an additive invariant defined on real algebraic sets.Suppose that for every pairX;Y of Nash diffeomorphic nonsingular compactreal algebraic sets we havee.X /D e.Y /. Then there exists a unique group ho-momorphismhe WZŒt �!G such thateDheıˇ. If , moreover, e is multiplicativethenhe is a ring homomorphism.

PROOF. Defineh.tn/ D e.Rn/. We claim that the additive invariant'.X / Dh.ˇ.X //� e.X / vanishes for every real algebraic setX . This is the case forX DR

n sinceˇ.Rn/D tn. By additivity, this is also the case forSnDRntpt .

By the existence of an algebraic compactification and resolution of singularities,it suffices to show the claim for compact nonsingular real algebraic sets.

Let X be a compact nonsingular real algebraic set and letQX be the blowup ofX along a smooth nowhere dense center. Then, using induction on dimX , wesee that'.X /D 0 if and only if '. QX /D 0. By the relative version of the Nash–Tognoli Theorem, the same result holds if we have thatQX is Nash diffeomorphicto the blowup of a nowhere dense Nash submanifold ofX . Thus the claim andhence the first statement follows from Mikhalkin’s Theorem. ˜

Following earlier results of Ax and Borel, K. Kurdyka showedin [20] thatany regular injective self-morphismf W X ! X of a real algebraic varietyis surjective. It was then showed in [29] that an injective continuous self-mapf WX !X of a locally compactAS set, such that the graph off is anAS set,is a homeomorphism. The arguments of both [20] and [29] are topological anduse the continuity off in essential way. The use of additive invariants allowsus to handle the non-continuous case.

THEOREM 4.6. Let X be anAS set and letf W X ! X be a map withAS

graph. If f is injective then it is surjective.

PROOF. It suffices to show that there exists a finite decompositionX DF

Xi

into locally compactAS sets such that for eachi , f restricted toXi is a home-omorphism onto its image. Then, by Corollary 4.3,

ˇ�

X nF

i f .Xi/�

D ˇ.X /�P

i ˇ.Xi/D 0;

and hence, by the degree property,X nF

i f .Xi/D?.To get the required decomposition first we note that by classical theory there

exists a semialgebraic stratification ofX DF

Sj such thatf restricted to eachstratum is real analytic. We show that we may choose strata belonging to theclassAS. (We do not require the strata to be connected.) By [20] and [29],each semialgebraic subsetA of a real algebraic varietyV has a minimalAS

closure inV , denotedAAS . Moreover ifA is AS then dimAAS nA < dimA.Therefore, we may take as the first subset of the decomposition the complement


in X of theAS closure of the union of strataSj of dimension< dimX , andthen proceed by induction on dimension.

Let X DF

Sj be a stratification withAS strata and such thatf is analyticon each stratum. Then, for each stratumSj , we apply the above argument tof �1 defined onf .Sj /. The induced subdivision off .Sj /, and hence ofSj ,satisfies the required property. ˜

Of course, in general, surjectivity does not apply injectivity for a self-map. Nev-ertheless we have the following result.

THEOREM 4.7. Let X be anAS set and letf W X ! X be a surjective mapwith AS graph. Suppose that there exist a finiteAS decompositionX D

F

Yi

andAS setsFi such that for eachi , f �1.Yi/ is homeomorphic toYi �Fi by ahomeomorphism withAS graph. Thenf is injective.

PROOF. We have

0D ˇ.X /�ˇ.f .X //DX

ˇ.Yi/.ˇ.Fi/� 1/:

Therefore .Fi/�1D 0 for eachi ; otherwise the polynomial on the right-handside would be nonzero with strictly positive leading coefficient. ˜

4B. Application to spaces of orderings.Let V be an irreducible real algebraicsubset ofRN . A function' W V ! Z is calledalgebraically constructibleif itsatisfies one of the following equivalent properties [24; 30]:

(i) There exist a finite family of proper regular morphismsfi W Zi ! V , andintegersmi , such that for allx 2 V ,

'.x/DX

i

mi�.f�1

i .x/\Zi/: (4-2)

(ii) There are finitely many polynomialsPi 2 RŒx1; : : : ;xN � such that for allx 2 V ,

'.x/DX

i

sgnPi.x/:

Let KDK.V / denote the field of rational functions ofV . A function' WV !Z

is generically algebraically constructible if and only if can be identified, up toa set of dimension smaller dimV , with the signature of a quadratic form overK. Denote byX the real spectrum ofK. A (semialgebraically) constructiblefunction onV , up to a set of dimension smaller dimV , can be identified witha continuous function' W X ! Z; see [5, Chapter 7], [23], and [6]. The repre-sentation theorem of Becker and Brocker gives a fan criterion for recognizinggenerically algebraically constructible function onV . The following two theo-rems are due to I. Bonnard.


THEOREM 4.8 [6]. A constructible function' W V ! Z is generically alge-braically constructible if and only for any finite fanF of X

X

�2F

'.�/� 0 mod jF j: (4-3)

For the notion of a fan see [5, Chapter 7], [23], and [6]. The number of elementsjF j of a finite fanF is always a power of2. It is known that for every finite fanF of X there exists a valuation ringBF of K compatible withF , and on whoseresidue field the fanF induces exactly one or two distinct orderings. Denoteby F the set of these fans ofK for which the residue field induces only oneordering.

THEOREM 4.9 [6]. A constructible function' W V ! Z is generically Nashconstructible if and only if(4-3) holds for every fanF 2 F .

The following question is due to M. Coste and M. A. Marshall [23, Question 2]:

Suppose that a constructible function' WV !Z satisfies(4-3) for every fanFof K with jF j � 2n. Does there exists a generically algebraically constructiblefunction W V ! Z such that for eachx 2 V , '.x/� .x/� 0 mod 2n?

We give a positive answer to the Nash constructible analog ofthis question.

THEOREM 4.10. Suppose that a constructible function' W V ! Z satisfies(4-3) for every fanF 2 F with jF j � 2n. Then there exists a generically Nashconstructible function W V ! Z such that for eachx 2 V , '.x/� .x/ � 0

mod 2n.

PROOF. We proceed by induction onn and onk D dimV . The casen D 0 istrivial.

Suppose' W V ! Z satisfies (4-3) for every fanF 2F with jF j � 2n, n� 1.By the inductive assumption,' is congruent modulo2n�1 to a generically Nashconstructible function n�1. By replacing' by ' � n�1, we may suppose2n�1 divides'.

We may also supposeV compact and nonsingular, just choosing a model forK DK.V /. Moreover, by resolution of singularities, we may assume that ' isconstant in the complement of a normal crossing divisorD D

S

Di � V .Let c be given by (3-6) with'c;p D ' andp D n�k �1. At a generic point

x of Di define@Di'.x/ as the average of the values of' on the local connected

components ofV n D at x. Then @c DP

i @ic, where@ic is described by@Di

' as in (3-7) (see [7]). Note that the constructible functions@Di' satisfy the

inductive assumption forn� 1. Hence each@Di' is congruent to a generically

Nash constructible function modulo2n�1. In other words@c 2 NpCk�1.V /.


Then by Corollary 3.12 we havec 2NpCk.V /, which implies the statement ofthe theorem. ˜

Using Corollary 3.12 we obtain the following result. The original proof wasbased on the fan criterion (Theorem 4.9).

PROPOSITION4.11 [7]. LetV �RN be compact, irreducible, and nonsingular.

Suppose that the constructible function' W V ! Z is locally constant in thecomplement of a normal crossing divisorDD

S

Di �V . Then' is genericallyNash constructible if and only if@D' is generically Nash constructible.

PROOF. We show only ((). Suppose2kCpj' generically, wherek D dimV ,and let c be given by (3-6) with'c;p D '. Then by our assumption@c 2NpCk�1.V /. By Corollary 3.12 we havec 2 NpCk.V /, which shows that,modulo2kCpC1, ' coincides with a generically Nash constructible function .Then we apply the same argument to' � . ˜

REMARK 4.12. We note that Proposition 4.11 implies neither Theorem4.10nor Corollary 3.12. Similarly the analog of this proposition proved in [6] doesnot give an answer to Coste and Marshall’s question.

5. The toric filtration

In their investigation of the relation between the homologyof the real andcomplex points of a toric variety [4], Bihanet al.define a filtration on the cellu-lar chain complex of a real toric variety. We prove that this filtered complexis quasi-isomorphic to the semialgebraic chain complex with the Nash con-structible filtration. Thus the toric filtered chain complexrealizes the weightcomplex, and the real toric spectral sequence of [4] is isomorphic to the weightspectral sequence.

For background on toric varieties see [12]. We use a simplified version ofthe notation of [4]. Let� be a rational fan inRn, and letX� be the real toricvariety defined by�. The groupTD .R�/n acts onX�, and thek-dimensionalorbitsO� of this action correspond to the codimensionk cones� of �.

The positive partX C

�of X� is a closed semialgebraic subset ofX�, and there

is a canonical retractionr WX�!X C

�that can be identified with the orbit map

of the action of the finite groupT D .S0/n onX�, whereS0Df�1;C1g�R�.

TheT -quotient of thek-dimensionalT-orbitO� is a semialgebraick-cell c� ofX C

�, andO� is a disjoint union ofk-cells, each of which maps homeomorphi-

cally ontoc� by the quotient map. This decomposition defines a cell structure onX� such thatX C

�is a subcomplex and the quotient map is cellular. LetC�.�/

be the cellular chain complex ofX� with coefficients inZ2. The closures ofthe cells ofX� are not necessarily compact, but they are semialgebraic subsets


of X�. Thus we have a chain map

˛ W C�.�/! C�.X�/ (5-1)

from cellular chains to semialgebraic chains.Thetoric filtration of the cellular chain complexC�.�/ is defined as follows

[4]. For eachk � 0 we define vector subspaces

0D T�k�1Ck.�/� T�kCk.�/� T�kC1Ck.�/� � � � � T0Ck.�/D Ck.�/;

(5-2)such [email protected].�//� TpCk�1.�/ for all k andp.

Let � be a cone of the fan�, with codim� D k. Let Ck.�/ be the subspaceof Ck.�/ spanned by thek-cells ofO� . Then

Ck.�/DM

codim� D k

Ck.�/:

The orbitO� has a distinguished pointx� 2 c� �X C

�. LetT� DT=T x� , where

T x� is theT -stabilizer ofx� . We identify the orbitT �x� with the multiplicativegroupT� . Eachk-cell ofO� contains a unique point of the orbitT �x� . Thus wecan make the identificationCk.�/D C0.T� /, the set of formal sums

P

i ai Œgi �,whereai 2 Z2 andgi 2 T� . The multiplication ofT� defines a multiplicationon C0.T� /, so thatC0.T� / is just the group algebra ofT� overZ2 .

Let I� be the augmentation ideal of the algebraC0.T� /, that is,

I� D KerŒ" W C0.T� /! Z2� with "P

i

ai Œgi �DP

i

ai :

For p � 0 we defineTpCk.�/ to be the subspace corresponding to the ideal.I� /

�p � C0.T� /, and we let

TpCk.�/DX

codim� D k

TpCk.�/:

If � < � in � and codim� D codim� � 1, the geometry of� determines agroup homomorphism'�� W T� ! T� (see [4]). Let@�� W Ck.�/! Ck�1.�/ bethe induced algebra homomorphism. We have@�� .I� /�I� . The boundary map@k W Ck.�/! Ck�1.�/ is given by@k.�/D

P

� @�� .�/, [email protected].�// �

TpCk�1.�/, soTpC�.�/ is a subcomplex ofC�.�/.

PROPOSITION5.1. For all k � 0 andp � 0, the chain map (5-1) takes thetoric filtration (5-2) to the Nash filtration(3-1),

˛.TpCk.�//�NpCk.X�/:


PROOF. It suffices to show that for every cone� 2� with codim� D k,

˛.TpCk.�//�NpCk.O� /:

The varietyO� is isomorphic to.R�/k , the toric variety of the trivial fanf0g inR

k , and the action ofT� onO� corresponds to the action ofTkDf�1;C1gk on.R�/k . Thek-cells of .R�/k are its connected components. LetIk � C0.Tk/

be the augmentation ideal. LetqD�p, so0� q � k. The vector spaceC0.Tk/

has dimension2k , and for eachq the quotientIq=IqC1 has dimension�

kq

�

.A basis forIq=IqC1 can be defined as follows. Lett1; : : : ; tk be the standardgenerators of the multiplicative groupTk ,

ti D .ti1; : : : ; tik/; tij D

�

�1 if i D j ,1 if i ¤ j .

If S � f1; : : : ; kg, let TS be the subgroup ofTk generated byfti I i 2 Sg,and defineŒTS � 2 C0.Tk/ by

ŒTS �DX

t2TS

Œt �:

ThenfŒTS � I jS j D qg is a basis forIq=IqC1 (see [4]).To prove that ..Ik/

q/�N�qCk..R�/k/we just need to show that ifjS jDq

then˛.ŒTS �/ 2N�qCk..R�/k/. Now the chain .ŒTS �/ 2 Ck..R

�/k/ is repre-sented by the semialgebraic setAS � .R

�/k ,

AS D f.x1; : : : ;xk/ I xi > 0; i … Sg;

and' D 2k�q1ASis Nash constructible. To see this consider the compactifi-

cation .P1.R//k of .R�/k . We have' D Q'j.R�/k , where Q' D f�1.P1.R//k ,

with f W .P1.R//k ! .P1.R//k defined as follows. Ifz D .u W v/ 2 P1.R/, letf1.z/D .u W v/, andf2.z/D .u

2 W v2/. Then

f .z1; : : : ; zk/D .w1; : : : ; wk/; wi D

�

f1.zi/ if i 2 S ,f2.zi/ if i … S .

This completes the proof. ˜

LEMMA 5.2. Let� be a codimensionk cone of�, and let

Ci.�/D

�

Ck.�/ if i D k,0 if i ¤ k.

For all p � 0,˛� WH�.TpC�.�//!H�.NpC�.O� //

is an isomorphism.


PROOF. Again we only need to consider the caseO� D .R�/k , where� is the

trivial cone0 in Rn. Now

Hi.C�.0//D

�

Ck.0/ if i D k,0 if i ¤ k,

and

Hi.C�..R�/k//D

�

Ker@k if i D k,0 if i ¤ k,

where@k W Ck..R�/k/! Ck�1..R

�/k/. The vector space Ker@k has basis thecycles represented by the components of.R�/k , and˛ W Ck.0/! Ck..R

�/k/

is a bijection from the cells ofCk.0/ to the components of.R�/k . Thus˛ WCk.0/ ! Ker@k is an isomorphism of vector spaces. Therefore˛ takes thebasisfAS I jS j D qgqD0;:::;k to a basis of Ker@k . The proof of Proposition5.1 shows that ifjS j � q thenAS 2 N�qCk..R

�/k/. We claim further that ifjS j< q thenAS …N�qCk..R

�/k/. It follows thatfAS I jS j � qg is a basis forHk.N�qC�..R

�/k/, and so

˛� WH�.T�qC�.0//!H�.N�qC�..R�/k//

is an isomorphism, as desired.To prove the claim, it suffices to show that ifAS is the closure ofAS in R

n,thenAS … N�qCk..R

�/k/. We show this by induction onk. The casek D 1

is clear: IfAD fx I x � 0g thenA … N�1C1.R/ because@A ¤ 0. In generalASDf.x1; : : : ;xk/ I xi�0; i …Sg. SupposeAS is .�q/-Nash constructible forsomeq> jS j. Then there exists' WRk! 2k�qZ generically Nash constructiblein dimensionk such thatAS D fx 2 R

k I '.x/ … 2k�qC1Zg; up to a set ofdimension< k. Let j …S , and letWj Df.x1; : : : ;xk/ I xj D 0gŠR

k�1. Then@Wj

' WWj ! 2k�q�1Z, andAS \Wj D fx 2Wj I @Wj'.x/ … 2k�qZg, up to

a set of dimension< k � 1. HenceAS \Wj 2N�qCk�1.Wj /. But

AS \Wj D f.x; : : : ;xk/ I xj D 0;xi � 0; i … Sg;

and so by the inductive hypothesisAS \Wj …N�qCk�1.Wj /, which is a con-tradiction. ˜

LEMMA 5.3. For every toric varietyX� and everyp � 0,

˛� WH�.TpC�.�//!H�.NpC�.X�//

is an isomorphism.

PROOF. We show by induction on orbits that the lemma is true for every varietyZ that is a union of orbits in the toric varietyX�. Let˙ be a subset of�, andlet˙ 0 D˙ n f�g, where� 2˙ is a minimal cone,i. e. there is no� 2˙ with� < � . Let Z andZ0 be the unions of the orbits corresponding to cones in˙


and˙ 0, respectively. ThenZ0 is closed inZ, andZ nZ0 D O� . We have acommutative diagram with exact rows:

� � � �! Hi.TpC�.˙0// �! Hi.TpC�.˙// �! Hi.TpC�.�// �! Hi�1.TpC�.˙

0// �! � � �?

?

?

y

ˇi

?

?

?

y

i

?

?

?

y

˛i

?

?

?

y

ˇi�1

� � � �! Hi.NpC�.˙0// �! Hi.NpC�.˙// �! Hi.NpC�.�// �! Hi�1.NpC�.˙

0// �! � � �

By Lemma 5.3 i is an isomorphism for alli . By inductive hypothesis i is anisomorphism for alli . Therefore i is an isomorphism for alli . ˜

THEOREM 5.4. For every toric varietyX� and everyp � 0,

˛� WH�

�

TpC�.�/

Tp�1C�.�/

�

!H�

�

NpC�.X�/

Np�1C�.X�/

�

is an isomorphism.

PROOF. This follows from Lemma 5.3 and the long exact homology sequencesof the pairs.TpC�.�/; Tp�1C�.�// and.NpC�.X�/;Np�1C�.X�//. ˜

Thus for every toric varietyX� the toric filtered complexT C�.�/ is quasi-isomorphic to the Nash constructible filtered complexNC�.X�/, and so thetoric spectral sequence [4] is isomorphic to the weight spectral sequence.

EXAMPLE 5.5. For toric varieties of dimension at most 4, the toric spectralsequence collapses [4; 35]. V. Hower [17] discovered that the spectral sequencedoes not collapse for the 6-dimensional projective toric variety associated to thematroid of the Fano plane.

Appendix: Semialgebraic chains

In this appendix we denote byX a locally compact semialgebraic set (i.e.a semialgebraic subset of the set of real points of a real algebraic variety) andby C�.X / the complex of semialgebraic chains ofX with closed supports andcoefficients inZ2. The complexC�.X / has the following geometric description,which is equivalent to the usual definition using a semialgebraic triangulation[5, 11.7].

A semialgebraic chainc of X is an equivalence class of closed semialgebraicsubsets ofX . For k � 0, let Sk.X / be theZ2 vector space generated by theclosed semialgebraic subsets ofX of dimension� k. ThenCk.X / is theZ2

vector space obtained as the quotient ofSk.X / by the following relations:

(i) If A andB are closed semialgebraic subsets ofX of dimension at mostk,then

ACB � cl.A�B/;


whereA�B D .A[B/ n .A\B/ is the symmetric difference ofA andB,and cl denotes closure.

(ii) If A is a closed semialgebraic subset ofX and dimA< k, thenA� 0.

If the chainc is represented by the semialgebraic setA, we write c D ŒA�. Ifc2Ck.X /, thesupportof c, denoted Suppc, is the smallest closed semialgebraicset representingc. If c D ŒA� then Suppc D fx 2A I dimx AD kg.

The boundaryoperator@k W Ck.X / ! Ck�1.X / can be defined using thelink operator� on constructible functions [24]. Ifc 2Ck.X / with cD ŒA�, then@kc D Œ@A�, where@A D fx 2 A I �1A.x/� 1 .mod 2/g. The operator@k iswell-defined, and@k�1@k D 0, since� ı�D 2�.

If f W X ! Y is a proper continuous semialgebraic map, thepushforwardhomomorphismf� W Ck.X /! Ck.Y / is defined as follows. LetA be a repre-sentative ofc. Thenf .A/ � B1C � � � CBl , where each closed semialgebraicsetBi has the property that #.A\ f �1.y// is constant mod 2 onBi nB0

i forsome closed semialgebraic setBi �Bi with dimB0

i < k. For eachi let ni 2Z2

be this constant value. Thenf�.c/D n1ŒB1�C � � �C nl ŒBl �.Alternately,f�.c/ D ŒB�, whereB D clfy 2 Y I f�1A.y/ � 1 .mod 2/g,

andf� is pushforward for constructible functions [24]. From thisdefinition itis easy to prove the standard propertiesg�f� D .gf /� and@kf� D f�@k .

We use two basic operations on semialgebraic chains: restriction and closure.These operations do not commute with the boundary operator in general.

Let c 2 Ck.X / and letZ � X be a locally closed semialgebraic subset. Ifc D ŒA�, we define therestriction by cjZ D ŒA\Z� 2 Ck.Z/. This operationis well-defined. IfU is an open semialgebraic subset ofX , [email protected] / D

.@kc/jU .Now let c 2 Ck.Z/ with Z � X locally closed semialgebraic. Ifc D ŒA�

we define theclosureby Nc D Œcl.A/� 2 Ck.X /, where cl.A/ is the closure ofAin X . Closure is a well-defined operation on semialgebraic chains.

By means of the restriction and closure operations, we definethe pullback of achain in the following situation, which can be applied to an acyclic square (1-2)of real algebraic varieties. Consider a square of locally closed semialgebraicsets,

QY ��! QX?

?

y

?

?

y

�

Yi

��! X

such that� W QX!X is a proper continuous semialgebraic map,i is the inclusionof a closed semialgebraic subset,QY D ��1.Y /, and the restriction of� is ahomeomorphism� 0 W QX n QY ! X nY . Let c 2 Ck.X /. We define thepullback


��1c 2 Ck. QX / by the formula

��1c D ..� 0/�1/�.cjX nY /:

Pullback does not commute with the boundary operator in general.

Acknowledgement

We thank Michel Coste for comments on a preliminary version of this paper.

References

[1] D. Abramovich, K. Karu, K. Matsuki, J. Włodarczyk,Torification and factorizationof birational maps, J. Amer. Math. Soc.29 (2002), 531–572.

[2] S. Akbulut, H. King,The topology of real algebraic sets, Enseign. Math.29 (1983),221–261.

[3] S. Akbulut, H. King,Topology of Real Algebraic Sets, MSRI Publ.25, Springer,New York, 1992.

[4] F. Bihan, M. Franz, C. McCrory, J. van Hamel,Is every toric variety an M-variety?,Manuscripta Math.120(2006), 217–232.

[5] J. Bochnak, M. Coste, M.-F. Roy,Real Algebraic Geometry, Springer, New York,1992.

[6] I. Bonnard,Un critere pour reconaitre les fonctions algebriquement constructibles,J. Reine Angew. Math.526(2000), 61–88.

[7] I. Bonnard,Nash constructible functions, Manuscripta Math.112(2003), 55–75.

[8] P. Deligne,Theorie de Hodge II, IHES Publ. Math.40 (1971), 5–58.

[9] P. Deligne,Theorie de Hodge III, IHES Publ. Math.44 (1974), 5–77.

[10] P. Deligne,Poids dans la cohomologie des varietes algebriques, Proc. Int. Cong.Math. Vancouver (1974), 79–85.

[11] G. Fichou,Motivic invariants of arc-symmetric sets and blow-Nash equivalence,Compositio Math.141(2005) 655–688.

[12] W. Fulton,Introduction to Toric Varieties, Annals of Math. Studies131, Princeton,1993.

[13] A. Gabrielov, N. Vorobjov, T. Zell,Betti numbers of semialgebraic and sub-Pfaffian sets, J. London Math. Soc. (2)69 (2004), 27–43.

[14] H. Gillet, C. Soule, Descent, motives, and K-theory, J. Reine Angew. Math.478(1996), 127–176.

[15] F. Guillen, V. Navarro Aznar,Un critere d’extension des foncteurs definis sur lesschemas lisses, IHES Publ. Math.95 (2002), 1–83.

[16] F. Guillen, V. Navarro Aznar,Cohomological descent and weight filtration(2003).(Abstract: http://congreso.us.es/rsme-ams/sesionpdf/sesion13.pdf.)


[17] V. Hower, A counterexample to the maximality of toric varieties, Proc. Amer.Math. Soc.,136(2008), 4139–4142.

[18] W. Kucharz,Homology classes represented by semialgebraic arc-symmetric sets,Bull. London Math. Soc.37 (2005), 514–524.

[19] K. Kurdyka,Ensembles semi-algebriques symetriques par arcs, Math. Ann. 281(1988), 445–462.

[20] K. Kurdyka, Injective endomorphisms of real algebraic sets are surjective, Math.Ann. 313no.1 (1999), 69–83

[21] K. Kurdyka, A. Parusinski,Arc-symmetric sets and arc-analytic mappings, Panora-mas & Syntheses24, Soc. Math. France (2007), 33–67.

[22] S. MacLane,Homology, Springer, Berlin 1963.

[23] M. A. Marshall,Open questions in the theory of spaces of orderings, J. SymbolicLogic 67 (2002), no. 1, 341–352.

[24] C. McCrory, A. Parusinski, Algebraically constructible functions, Ann. Sci.Ec.Norm. Sup.30 (1997), 527–552.

[25] C. McCrory, A. Parusinski, Virtual Betti numbers of real algebraic varieties,Comptes Rendus Acad. Sci. Paris, Ser. I,336(2003), 763–768.(See also http://arxiv.org/pdf/math.AG/0210374.)

[26] G. Mikhalkin, Blowup equivalence of smooth closed manifolds, Topology 36(1997), 287–299.

[27] G. Mikhalkin, Birational equivalence for smooth manifolds with boundary, Al-gebra i Analiz 11 (1999), no. 5, 152–165. In Russian; translation in St. PetersburgMath. J. 11 (2000), no. 5, 827–836

[28] M. Nagata,Imbedding of an abstract variety in a complete variety, J. Math. KyotoU. 2 (1962), 1–10.

[29] A. Parusinski, Topology of injective endomorphisms of real algebraic sets, Math.Ann. 328(2004), 353–372.

[30] A. Parusinski, Z. Szafraniec,Algebraically constructible functions and signs ofpolynomials, Manuscripta Math.93 (1997), no. 4, 443–456.

[31] H. Pennaneac’h,Algebraically constructible chains, Ann. Inst. Fourier (Grenoble)51 (2001), no. 4, 939–994,

[32] H. Pennaneac’h,Nash constructible chains, preprint Universita di Pisa, (2003).

[33] H. Pennaneac’h,Virtual and non-virtual algebraic Betti numbers, Adv. Geom.5(2005), no. 2, 187–193.

[34] C. Peters, J. Steenbrink,Mixed Hodge Structures, Springer, Berlin, 2008.

[35] A. Sine, Probleme de maximalite pour les varietes toriques, These Doctorale,Universite d’Angers 2007.

[36] R. Thom,Quelques proprietes globales des varietes differentiables, Comm. Math.Helv. 28 (1954), 17–86.


[37] B. Totaro,Topology of singular algebraic varieties, Proc. Int. Cong. Math. Beijing(2002), 533-541.

CLINT MCCRORY

MATHEMATICS DEPARTMENT

UNIVERSITY OF GEORGIA

ATHENS, GA 30602UNITED STATES

[email protected]

ADAM PARUSINSKI

LABORATOIRE J.-A. DIEUDONNE

U.M.R. No 6621DU C.N.R.S.UNIVERSITE DE NICE - SOPHIA ANTIPOLIS

PARC VALROSE

06108 NICE CEDEX 02FRANCE

[email protected]


On Milnor classesof complex hypersurfaces

LAURENTIU MAXIM

ABSTRACT. We revisit known results about the Milnor class of a singularcomplex hypersurface, and rephrase some of them in a way thatallows fora better comparison with the topological formula of Cappelland Shaneson fortheL-class of such a hypersurface. Our approach is based on Verdier’s special-ization property for the Chern–MacPherson class, and simple constructiblefunction calculus.

1. Introduction

It is well-known that for a compact complex hypersurfaceX with only iso-lated singularities the sum of the Milnor numbers at the singular points measures(up to a sign) the difference between the topological Euler characteristic ofXand that of a nonsingular hypersurface linearly equivalentto X , provided sucha hypersurface exists. This led Parusinski to a generalization of the notion ofMilnor number to nonisolated hypersurface singularities [16], which in the caseof isolated singularities reduces to the sum of Milnor numbers at the singularpoints.

For a (possibly singular) compact complex hypersurfaceX , the Euler char-acteristic�.X / equals the degree of the zero-dimensional component of theChern–MacPherson homology classc�.X /; see [15]. On the other hand, theEuler characteristic of a nonsingular hypersurface linearly equivalent toX isjust the degree of the Poincare dual of the Chern class of the virtual tangentbundle ofX , that is, the degree of the Fulton–Johnson classcFJ

�.X / [10; 11].

Thus, Parusinski’s Milnor number equals (up to a sign) the degree of the ho-mology classcFJ

�.X /� c�.X /. It is therefore natural to try to understand the

This work was partially supported by NSF and a PSC-CUNY Research Award.

161

162 LAURENTIU MAXIM

higher-degree components of this difference class, which usually is called theMilnor classof X . The study of the Milnor class also comes up naturally whilesearching for a Verdier-type Riemann–Roch theorem for the Chern–MacPhersonclasses (see [20; 22; 23]); indeed, the Milnor class measures the defect of com-mutativity in a Verdier–Riemann–Roch diagram for MacPherson’s Chern classtransformation.

While the problem of understanding the Milnor class in termsof invariants ofsingularities can be formulated in more general contexts (e.g., for local completeintersections, or regular embeddings in arbitrary codimension, see [19; 20]), inthis note we restrict ourselves, for simplicity, only to thecase of hypersurfaces(i.e., regular embeddings in codimension1) in complex manifolds. We recallknown results about the Milnor class of a singular hypersurface, and rephrasesome of these results in a way that, we believe, reflects better the geometry ofthe singular locus in terms of its stratification. For more comprehensive surveyson Milnor classes, the interested reader is advised to consult [2; 3; 17; 22].

The approach presented in this note is based on a well-known specializationargument [21], and simple calculus of constructible functions as developed in[9]. While this approach is not new (see [18; 19; 20] for similar considerations),the formulation of our main results (Theorem 4.3, Corollary4.4 and Theo-rem 4.6) has the advantage of being conceptually very simple, and it allows for abetter comparison with the topological formula of Cappell and Shaneson [7; 8]for the L-classes of singular hypersurfaces. Indeed, we also explore a Chern-class analogue of Goresky–MacPherson’s homologyL-class [12], defined viathe constructible function associated to the intersectionchain complex of a va-riety (see [9]). This class, which for a varietyX is denoted byIc�.X /, encodesvery detailed information about the geometry of a fixed Whitney stratificationof X . In the case of hypersurfaces, we compare this class with theFulton–Johnson class, and derive a formula for their difference in terms of invariants ofthe singular locus.

2. Canonical bases for the group of constructible functions

Let X be a topological space with a finite partitionV into a disjoint union offinitely many connected subsetsV satisfying thefrontier condition:

W \V ¤? ÷ W � V :

The main examples of such spaces are complex algebraic or compact complexanalytic varieties with a fixed Whitney stratification. Consider onV the partialorder given by

W � V ” W � V :

ON MILNOR CLASSES OF COMPLEX HYPERSURFACES 163

We also writeW < V if W � V andW ¤ V .Let FV.X / be the abelian group ofV-constructible functions onX , that is,

functions˛ W X ! Z such that jV is constant for allV 2 V . This is a freeabelian group with basis

B1 WD f 1V j V 2 V g;

so that any 2 FV.X / can be written as

˛ DX

V 2V

˛.V / � 1V : (2-1)

In what follows, we will discuss two more canonical bases onFV.X /, see[9] for complete details. First, the collection

B2 WD f 1V j V 2 V g

is also a basis forFV.X /, since

1V DX

W �V

1W

and the transition matrixA D .aW ;V /, whereaW ;V is defined as1 if W � V

and0 otherwise, is upper triangular with respect to�, with all diagonal entriesequal to1 (soA is invertible). In this basis, a constructible function˛ 2FV.X /

can be expressed by the identity

˛ DX

V

˛.V / � O1V ; (2-2)

(see [9, Proposition 2.1]), where for eachV 2 V , we defineO1V inductively bythe formula

O1V D 1V �X

W <V

O1W :

Note that if there is a stratumS 2V which is dense inX , i.e.,S DX , soV �S

for all V 2 V , then (2-2) can be rewritten as

˛ D ˛.S/ � 1X CX

V<S

.˛.V /�˛.S// � O1V : (2-3)

If moreover˛jS D 0, this reduces further to

˛ DX

V<S

˛.V / � O1V : (2-4)

In order to describe the third basis for the group of constructible functions,assume moreover thatX is a topological pseudomanifold with a stratificationV by finitely many oriented strata ofevendimension. Then, by definition, the

164 LAURENTIU MAXIM

strata ofV satisfy the frontier condition, andV is locally topologically trivialalong each stratumV , with fibers the cone on a compact pseudomanifoldLV;X ,the link of V in X . Each stratumV , and also its closureV , get an inducedstratification of the same type. Important examples are provided by a complexalgebraic (or analytic) Whitney stratification of a reducedcomplex algebraic (orcompact complex analytic) variety.

For eachV 2 V , let ICV be the intersection cohomology complex [13] asso-ciated to the closure ofV in X . This is aV-constructible complex of sheaves(i.e., the restrictions of its cohomology sheaves to strataW < V are locallyconstant), satisfying the normalization property thatICV jV D QV (followingBorel’s indexing conventions). After extending by zero, weregard all theseintersection chain sheaves as complexes onX . Let us fix for eachW 2V a pointw 2W with inclusioniw W fwg Œ X . We now define a constructible functionicV 2FV.X / by taking stalkwise the Euler characteristic for the complex ICV .That is, forw 2W < V we let

icV .w/ WD �.i�

wICV /D �.IH �.cıLW ;V //defD I�.cıLW ;V /; (2-5)

wherecıLW ;V denotes the open cone on the linkLW ;V of W in V , andI�.�/

stands for the intersection homology Euler characteristic. Moreover,

icV jV D 1V : (2-6)

Since clearly supp.icV /D V , it is now easy to see that the collection

B3 WD f icV j V 2 V g

is another distinguished basis ofFV.X /. Indeed, by (2-6), the transition matrixto the basisf1V g is upper triangular with respect to�, with all diagonal entriesequal to1, so it is invertible. The advantage of working with the latter basis isthat it carries more information about the geometry of the chosen stratification.

Now assume thatX has an open dense stratumS 2 V so thatV � S forall V 2 V , e.g.,X is an irreducible reduced complex algebraic (resp., compactcomplex analytic) variety. For eachV 2 V n fSg define inductively

bic.V / WD icV �X

W <V

bic.W / � I�.cıLW ;V / 2 FV.X /: (2-7)

Then anyV-constructible function 2 FV.X / can be represented with respectto the basisficV jV 2 Vg by the following identity (see [9, Theorem 3.1]):

˛ D ˛.s/ � icX CX

V<S

�

˛.v/�˛.s/ � I�.cıLV;Y /�

� bic.V /: (2-8)


In the particular case whenjS D 0, i.e., supp.˛/ � X nS , this reduces to theidentity

˛ DX

V<S

˛.V / � bic.V /; (2-9)

which will become very important in the context of computingMilnor classesof singular complex hypersurfaces. Also, if we plug˛ D 1X in equation (2-8),we obtain under the assumptions in this paragraph the following comparisonformula (also valid if we replaceX by the closure of any given stratum ofV):

1X D icX CX

V<S

�

1� I�.cıLV;Y /�

� bic.V /: (2-10)

3. Chern classes of singular varieties

For the rest of the paper we specialize to the complex algebraic (respectively,compact complex analytic) context, withX a reduced complex algebraic (resp.,compact complex analytic) variety. There are several generalizations of the(total) Chern class of complex manifolds to the context of such singular va-rieties. Among these we mention here the Chern–MacPherson class [15] andthe Fulton–Johnson class [10; 11]. Both coincide with the Poincare dual of theChern class if the variety is smooth.

3.1. The Chern–MacPherson class. The groupFc.X / of complex alge-braically (resp., analytically) constructible functionsis defined as the directlimit of groups FV.X /, with respect to the directed systemfVg of Whitneystratifications ofX . Moreover, there is a functorial pushdown transformationof constructible functions, namely, a proper complex algebraic (resp., analytic)mapf WX ! Y induces a group homomorphism

f� W Fc.X /! Fc.Y /;

defined by

f�.˛/.y/ WD �.˛jf �1.y//;

for � WFc.X /!Z the constructible function which for a closed algebraic (resp.,analytic) subspaceZ of X is given by

�.1Z / WD �.H�.Z//D �.Z/:

In particular, for such a closed subsetZ �X we have that

f�.1Z /.y/D �.Z\f�1.y//:

The fact that the pushdownf� is well-defined requires a stratification of themorphismf (see [15]).

166 LAURENTIU MAXIM

The Chern class transformation of MacPherson [15] is the group homomor-phism

c� W Fc.X /!H BM2�

.X IZ/;

which commutes with proper pushdowns, and is uniquely characterized by thisproperty together with the normalization axiom asserting that

c�.1X /D c�.TX /\ ŒX �

if X is a complex algebraic (resp., analytic) manifold. Herec�.TX / is theChern cohomology class of the tangent bundleTX . Also H BM

2�.�/ stands for

the even-dimensional Borel–Moore homology. The Chern–MacPherson classof X is then defined as

c�.X / WD c�.1X / 2H BM2�

.X IZ/:

If X is compact, the degree ofc�.X / is just�.X /, the topological Euler char-acteristic ofX . Similarly, we set

Ic�.X / WD c�.icX /;

which is another possible extension of Chern classes of manifolds to the singularsetting. Of course, ifX is smooth thenc�.X / D Ic�.X /, but in general theydiffer for singular varieties, their difference being a measure of the singularlocus, which, moreover, is computable in terms of the geometry of the stratifi-cation. Indeed, by applyingc� to the identity (2-10), we obtain the followingcomparison formula:

c�.X /� Ic�.X /DX

V<S

�

1� I�.cıLV;Y /�

� bIc�.V /: (3-1)

If X is compact, the degree ofIc�.X / is justI�.X /, the intersection homologyEuler characteristic ofX .

3.2. The Fulton–Johnson class.Assume thatX is a local complete intersectionembedded in a complex manifoldM with inclusion

XiŒM:

If NX M denotes the normal cone ofX in M , then thevirtual tangent bundleof X , that is,

TvirX WD Œ i�TM �NX M � 2K0.X /; (3-2)

is a well-defined element in the Grothendieck group of vectorbundles onX

(e.g., see [11][Ex.4.2.6]), so one can associate to the pair.M;X / an intrinsichomology class,cFJ

� .X / 2 H BM2�

.X IZ/, called the Fulton–Johnson class anddefined as follows (see [10; 11]):

cFJ� .X / WD c�.TvirX /\ ŒX �: (3-3)


Of course, ifX is also smooth, thenTvirX coincides with the (class of the)usual tangent bundle ofX , andcFJ

�.X / is in this case just the Poincare dual of

c�.TX /.

4. Milnor classes of hypersurfaces

This section is devoted to comparing the two notions of Chernclasses men-tioned in the previous section. For simplicity, we restrictto the case whenX is a hypersurface in a complex manifoldM . As already mentioned, theChern–MacPherson class and the Fulton–Johnson class coincide if X is smooth.However, they differ in the singular case. For example, ifX has only isolatedsingularities, the difference is (up to a sign) the sum of theMilnor numbersattached to the singular points. For this reason, the differencecFJ

� .X /� c�.X /

is usually called theMilnor classof X , and is denoted byM.X /.1 The Milnorclass is a homology class supported on the singular locus ofX , and it has beenrecently studied by many authors using quite different methods, e.g., see [1; 2;3; 4; 5; 6; 18; 17; 19; 20; 22]. For example, it was computed in [18] (see also[17; 22]) as a weighted sum in the Chern–MacPherson classes of closures ofsingular strata ofX , the weights depending only on the normal information tothe strata. The approach we follow here is that of [19; 20], and relies only onthe simple calculus of constructible functions, as outlined in Section 2, togetherwith a well-known specialization argument due to Verdier [21].

Assume in what follows thatX is a reduced complex analytic hypersurface,which is globally defined as the zero-set of a holomorphic functionf WM !D

with a critical value at0 2 D, for M a compact complex manifold andD theopen unit disc about0 2 C. For each pointx 2 X , we have a correspondingMilnor fibration with fiber

Mf;x WD Bı.x/\f�1.t/

for appropriate choices of0< jt j � ı� 1.Denote byL the trivial line bundle onM , obtained by pulling back byf

the tangent bundle ofC. Then the virtual tangent bundle ofX can be identifiedwith

TvirX D ŒTM jX �LjX �: (4-1)

For eacht ¤ 0 small enough, each fiberXt WD f�1.t/ is a compact complex

manifold. Moreover, by compactness, given a regular neighborhoodU of X inM , there is a sufficiently smallt so thatXt �U . Denote byit the corresponding

1The definition of the Milnor class usually includes a sign, but for simplicity we choose to ignore it here.

168 LAURENTIU MAXIM

inclusion map. Also, letr WU!X be the obvious deformation retract.Verdier’sspecialization mapin homology is then defined as the composition

H D r� ı it � WH�.Xt /!H�.X /: (4-2)

There is also a specialization map defined on the level of constructible func-tions [21],

CF W Fc.M /! Fc.X /; (4-3)

which is just the constructible function version of Deligne’s nearby cycle functorfor constructible complexes of sheaves. This is defined by the formula

CF .˛/.x/D �.˛ � 1Mf;x/: (4-4)

In particular, CF .1M /D �X 2 Fc.X /; (4-5)

where�X WX ! Z is the constructible function defined by the rule:

�X .x/ WD �.Mf;x/; (4-6)

for all x 2X . This definition justifies the analogy with the nearby cycle functordefined on the level of constructible complexes of sheaves.

Verdier’s specialization property for the Chern–MacPherson classes [21] as-serts that for any 2 Fc.M / we have:

H c�.˛jXt/D c�. CF .˛//: (4-7)

In particular, by letting D 1M and using (4-5), we have that

H c�.Xt /D c�.�X /: (4-8)

We can now state the following easy (known) consequence:

PROPOSITION4.1.M.X /D c�. Q�X /; (4-9)

where Q�X 2Fc.X / is the constructible function supported on the singular locusof X , whose value atx 2X is defined by the Euler characteristic of the reducedcohomology of the corresponding Milnor fiber, i.e.,

Q�X .x/ WD �. QH�.Mf;x//: (4-10)

PROOF. First note that, sinceXt is smooth,

H c�.Xt /D H cFJ�.Xt /D cFJ

�.X /; (4-11)

where the last equality follows from the fact that the homology specializationmap H carries (the dual of) the Chern classes ofTM jXt

andLjXtinto (the

dual of) the Chern classes ofTM jX andLjX , respectively [21].


On the other hand,

c�.X /D c�.�X /� c�. Q�X /; (4-12)

so the desired identity follows by combining (4-8) and (4-11). ˜

REMARK 4.2. Note thatQ�X is the constructible function analogue of Deligne’svanishing cycle functor defined on constructible sheaves. Indeed,

Q�X D �CF .1M /; (4-13)

where�CF WD CF � i�, for i� WFc.M /!Fc.X / the pullback (restriction) ofconstructible functions defined byi�.˛/ WD ˛ ı i .

We are now ready to prove the main result of this note:

THEOREM 4.3. Let M be a compact complex manifold, andX a reduced hy-persurface defined by the zero-set of a holomorphic functionf WM ! D witha critical value at the origin. Fix a Whitney stratificationV on X , and for eachstratumV 2 V fix a pointv 2 V with corresponding Milnor fiberMf;v. Thenthe Milnor class ofX , i.e., the class

M.X / WD cFJ�.X /� c�.X / 2H�.X /;

can be computed by the formula

M.X /DX

V 2VV �Sing.X /

�. QH �.Mf;v// ��

c�.V /� c�.V nV /�

DX

V 2VV �Sing.X /

�. QH �.Mf;v// � Oc�.V /;

where for a stratumV 2 V we let Oc�.V / be defined inductively as

Oc�.V / WD c�.V /�X

W <V

Oc�.W /:

If , moreover, X is irreducible and we letS denote the dense open stratum inX , then:

M.X /DX

V<S

�. QH �.Mf;v// � bIc�.V /; (4-14)

where for eachV 2 V , bIc�.V / is defined inductively by

bIc�.V / WD Ic�.V /�X

W <V

I�.cıLW ;V / � bIc�.W /;

170 LAURENTIU MAXIM

for LW ;V the link ofW in V .2

PROOF. Recall that by Proposition 4.1 we have:

M.X /D c�. Q�X /: (4-15)

Moreover, the functionQ�X WX!Z is constructible with respect to the WhitneystratificationV . Therefore, as in (2-1) and (2-2), we can write

Q�X DX

V 2V

Q�X .v/ � 1V DX

V 2V

Q�X .v/ � .1V � 1V nV /DX

V 2V

Q�X .v/ � O1V :

Since smooth points have contractible Milnor fibers, only strata contained in thesingular locus ofX contribute to the above sums. The first part of the theoremfollows from (4-15) by applying the Chern–MacPherson transformation c� tothe last two of the above equalities.

If X is irreducible with dense open stratumS , then as in (2-9) we can write

Q�X DX

V<S

Q�X .v/ � bic.V /:

By applyingc�, we obtain the desired identity (4-14) from (4-15). ˜

As a consequence, the Chern–MacPherson class and the Fulton–Johnson classcoincide in dimensions greater than the dimension of the singular locus. And itcan be seen from any of the above formulae that ifX has only isolated singu-larities, the Milnor class is (up to a sign) just the sum of theMilnor numbers atthe singular points.

By combining (3-1) and (4-14) we also obtain a comparison formula for theFulton–Johnson classcFJ

�.X / and the Chern classIc�.X / defined via the in-

tersection cohomology chain sheaf.

COROLLARY 4.4. If X as above is a reduced irreducible hypersurface withdense open stratumS , then

IM.X / WD cFJ�.X /� Ic�.X /D

X

V<S

bIc�.V / ��

�.Mf;v/� I�.cıLV;X /�

:

(4-16)

Note that by constructible function calculus, we have that

IM.X /D c�.fI�X /; (4-17)

for fI�X WX !Z theV-constructible function whose value atv 2V is given by

fI�X .v/D �.Mf;v/� I�.cıLV;X /: (4-18)

2By the functoriality ofc�, we can regard all classesc�.V /, Oc�.V / and OIc�.V / associated to a stratumV 2V as homology classes inH�.X /. This is the reason why we apply the Chern–MacPherson transforma-tion c� only to closed subvarieties ofX .


By its definition,fI�X is supported on the singular locus ofX , so (2-9) can beused directly to prove (4-16).

REMARK 4.5. Our formula (4-16) should be compared to the topological for-mula of Cappell and Shaneson [7; 8] for the Goresky–MacPherson L-class [12]of an irreducible reduced complex hypersurfaceX �M as above, namely,

L�.TvirX /�L�.X /DX

V<S

L�.V / � �.lk.V //; (4-19)

where�.lk.V //2Z is a certain signature invariant associated to the link pairofthe stratumV in .M;X /. HereL�.TvirX / WDL�.TvirX /\ ŒX �, with L� theL-polynomial of Hirzebruch [14] defined in terms of the power seriesx=tanh.x/.The comparison is motivated by the fact that theL-class of a singular varietyX is a topological invariant associated to the intersection cohomology complexof the variety. We should point out that the Cappell–Shaneson formula holds inmuch greater generality, namely for real codimension two PLembeddings witheven codimension strata, and its proof relies on powerful algebraic cobordismdecompositions of self-dual sheaves. However, we believe that in the context ofcomplex algebraic/analytic geometry, a simpler proof could be given by using aspecialization argument similar to the one presented here.

More generally, assume thati W X ŒM is a regular embedding in codimen-sion one of complex algebraic (resp., compact complex analytic) spaces withM smooth. ThenX is locally defined inM by one equationff D 0g, andthe specialization map CF W Fc.M / ! Fc.X / is still well-defined, as it isindependent of the chosen local equation forX . In particular, we still have that CF .1M /D�, whose value at a pointx 2X is given by the Euler characteristicof a local Milnor fiber atx. In other words, ifff D 0g is a defining equationfor X nearx, then

Q�X .x/ WD �. QH�.Mf;x//; (4-20)

for Mf;x the corresponding Milnor fiber. Then arguments similar to those usedin this section apply to this more general situation, and yield the following result(see [20, Corollary 0.2] for equation (4-21) below):

THEOREM 4.6. Let i W X ŒM be a regular embedding in codimension oneof complex algebraic(resp., compact complex analytic) spaces withM smooth.Then,

M.X /D c�.NX M /�1\ c�. Q�X /; (4-21)

with Q� the constructible function supported on the singular locusof X , whosevalue at a pointx 2X is given by the Euler characteristic of the reduced coho-mology of a local Milnor fiber atx. So, if we assumeX irreducible with dense

172 LAURENTIU MAXIM

open stratumS , then in the notations of Theorem4.3we get

M.X /DX

V<S

c�.NX M /�1\�

c�.V /� c�.V nV /�

� Q�X .v/

DX

V<S

c�.NX M /�1\ Oc�.V / � Q�X .v/

DX

V<S

c�.NX M /�1\ bIc�.V / � Q�X .v/:

Similar considerations apply toIM.X /. (Again, by functoriality, we regard allclasses defined on the closure of a given stratum as homology classes inX .)

We conclude this note by recalling some functoriality results for the Milnorclass of hypersurfaces (see [20; 24] for complete details).More precisely, weare concerned with the behavior of the Milnor class under a proper pushdown.Similar results were obtained in [9] for the Chern–MacPherson classesc�.�/

andIc�.�/, respectively.Let us consider the cartesian diagram

QXj

��! QM

f

?

?

y

?

?

y

�

Xi

��! M

with M and QM compact analytic manifolds, and� W QM !M a proper mor-phism. Also assume thati and j are regular closed embeddings of (local)codimension one, withM irreducible. Then it’s easy to see thatN QX

QM '

f �.NX M /. Therefore, by (4-21) and the projection formula, one has

f�M. QX /D f�.c�.N QX

QM /�1\ c�. Q� QX//

D f�.f�c�.NX M /�1\ c�. Q� QX

//

D c�.NX M /�1\f�c�. Q� QX/:

Next, by the functoriality ofc� and the definition ofQ� QXin (4-13) we obtain

f�c�. Q� QX/D c�f�. Q� QX

/D c�f��CF .1 QM/D c��CF .��.1 QM

//;

where the last identity follows by proper base change. Assume now that�(hence alsof ) is an Euler morphism, i.e., the Euler characteristics of all itsfibers are the same (e.g.,� is smooth), and denote this value by�f . Then��.1 QM

/D �f � 1M , and it follows in this case that

f�M. QX /D �f �M.X /: (4-22)


But in the case of a general morphism we have that

f�M. QX /D �f �M.X /C c�.NX M /�1\ c��CF .˛/; (4-23)

for ˛ WD ��.1 QM/��f �1M , with �f the Euler characteristic of the generic fiber

of �. Note that is supported on the critical locus of the morphism�.To this end, we note that the above considerations can also beused to study

the push-forward of the classIM. QX / in the case whenQX is pure-dimensionalandX is irreducible and reduced. Let us choose a stratificationV on X withdense open stratumS , so thatf�.1 QX

/; f�.ic QX/ 2 FV.X / (e.g., chooseQV and

V complex Whitney stratifications onQX and X , respectively, so thatf is astratified submersion, and1X ; ic QX

2 F QV.X /). Then, since

IM. QX /DM. QX /C .c�. QX /� Ic�. QX //;

a formula forf�IM. QX / can be derived by using (4-23), together with the for-mulae from [9, Propositions 3.4 and 3.6] for the push-forward of the Chernclassesc�. QX / andIc�. QX /, respectively. We leave the details as an exercise forthe interested reader. We only want to point out that for an Euler morphism(with smooth generic fiber), we obtain

f�IM. QX /

D�f � IM.X /CX

V<S

�

�f I�.cıLV;X /�I�.f �1.cıLV;X //�

� bic.V /; (4-24)

where�f is the Euler characteristic of the generic fiberF of f (and�).

Acknowledgements

I am grateful to Jorg Schurmann for many inspiring conversations on thissubject. I also thank Sylvain Cappell, Anatoly Libgober andJulius Shanesonfor constant encouragement and advice.

References

[1] Aluffi, P., Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc.351(1999), no. 10, 3989–4026.

[2] Aluffi, P., Characteristic classes of singular varieties, in Topics in cohomologicalstudies of algebraic varieties, 1–32, Birkhauser, Basel, 2005.

[3] Brasselet, J.-P.,From Chern classes to Milnor classes–a history of characteristicclasses for singular varieties, in Singularities (Sapporo 1998), 31–52, Adv. Stud.Pure Math., 29, Kinokuniya, Tokyo, 2000.

174 LAURENTIU MAXIM

[4] Brasselet, J.-P., Lehmann, D., Seade, J., Suwa, T.,Milnor numbers and classes oflocal complete intersections, Proc. Japan Acad. Ser. A Math. Sci.75 (1999), no. 10,179–183.

[5] Brasselet, J.-P., Lehmann, D., Seade, J., Suwa, T.,Milnor classes of local completeintersections, Trans. Amer. Math. Soc.354(2001), 1351–1371.

[6] Brasselet, J.-P., Seade, J., Suwa, T.,An explicit cycle representing the Fulton–Johnson class, I, Seminaire & Congres10 (2005), p. 21–38.

[7] Cappell, S., Shaneson, J.,Characteristic classes, singular embeddings, and inter-section homology, Proc. Natl. Acad. Sci. USA, Vol84 (1991), 3954–3956.

[8] Cappell, S., Shaneson, J.,Singular spaces, characteristic classes, and intersectionhomology, Ann. of Math.134(1991), 325–374.

[9] Cappell, S. E., Maxim, L. G., Shaneson, J. L.,Euler characteristics of algebraicvarieties, Comm. Pure Appl. Math.61 (2008), no. 3, 409–421.

[10] Fulton, W., Johnson, K.,Canonical classes on singular varieties, ManuscriptaMath 32 (1980), 381–389.

[11] Fulton, W.,Intersection theory, Second edition. Springer, Berlin, 1998.

[12] Goresky, M., MacPherson, R.,Intersection homology theory, Topology19 (1980),no. 2, 135–162.

[13] Goresky, M., MacPherson, R.,Intersection homology theory, II, Inv. Math. 72(1983), no. 2, 77–129.

[14] Hirzebruch, F.,Topological methods in algebraic geometry, Springer, New York,1966.

[15] MacPherson, R.,Chern classes for singular algebraic varieties, Ann. of Math. (2)100(1974), 423–432.

[16] Parusinski, A., A generalization of the Milnor number, Math. Ann.281 (1988),247–254.

[17] Parusinski, A., Characteristic classes of singular varieties, in Singularity theoryand its applications, 347–367, Adv. Stud. Pure Math., 43, Math. Soc. Japan, Tokyo,2006.

[18] Parusinski, A., Pragacz, P.,Characteristic classes of hypersurfaces and character-istic cycles, J. Algebraic Geom.10 (2001), no. 1, 63–79.

[19] Schurmann, J.,Lectures on characteristic classes of constructible functions, inTopics in cohomological studies of algebraic varieties, 175–201, Birkhauser, Basel,2005.

[20] Schurmann, J.,A generalized Verdier-type Riemann–Roch theorem for Chern–Schwartz–MacPherson classes, arXiv:math/0202175.

[21] Verdier, J.-L.,Specialisation des classes de Chern, Asterisque82–83, 149–159(1981).


[22] Yokura, S.,On characteristic classes of complete intersections, in Algebraicgeometry: Hirzebruch 70 (Warsaw, 1998), 349–369, Contemp.Math., 241, Amer.Math. Soc., Providence, RI, 1999.

[23] Yokura, S.,On a Verdier-type Riemann–Roch for Chern–Schwartz–MacPhersonclass, Topology and its Applications 94 (1999), no. 1–3, 315–327.

[24] Yokura, S.,An application of bivariant theory to Milnor classes, Topology and itsApplications 115 (2001), 43–61.

LAURENTIU MAXIM

LAURENTIU MAXIM


UNIVERSITY OF WISCONSIN-MADISON

480 LINCOLN DRIVE

MADISON, WI 53706-1388UNITED STATES

[email protected]


An introduction to intersection homology withgeneral perversity functions

GREG FRIEDMAN

ABSTRACT. We provide an expository survey of the different notions ofper-versity in intersection homology and how different perversities require differ-ent definitions of intersection homology theory itself. We trace the key ideasfrom the introduction of intersection homology by Goresky and MacPhersonthrough to the recent and ongoing work of the author and others.

CONTENTS

1. Introduction 1772. The original definition of intersection homology 1803. Goresky–MacPherson perversities 1854. Sheaf-theoretic intersection homology 1895. Subperversities and superperversities 1966. “Correcting” the definition of intersection chains 1997. General perversities 2018. Back to sheaf theory 2079. Recent and future applications of general perversities 211

10. Saralegi’s relative intersection chains 21511. Habegger and Saper’s codimension� c intersection homology theory 217Acknowledgments 219References 219

1. Introduction

When Goresky and MacPherson first introduced intersection homology [32],they required its perversity parameters to satisfy a fairlyrigid set of constraints.Their perversities were functions on the codimensions of strata, Np W Z�2! Z,

2000 Mathematics Subject Classification:Primary: 55N33, 57N80; Secondary: 55N45, 55N30, 57P10.Keywords: intersection homology, perversity, pseudomanifold, Poincare duality, Deligne sheaf, intersectionpairing.

177

178 GREG FRIEDMAN

satisfying

Np.2/D 0 and Np.k/� Np.kC 1/� Np.k/C 1:

These strict requirements were necessary for Goresky and MacPherson toachieve their initial goals for intersection homology: that the intersection ho-mology groupsI NpH�.X / should satisfy a generalized form of Poincare dualityfor stratified pseudomanifolds and that they should be topological invariants,i.e., they should be independent of the choice of stratification of X .

In the ensuing years, perversity parameters have evolved asthe applicationsof intersection homology have evolved, and in many cases thebasic definitionsof intersection homology itself have had to evolve as well. Today, there areimportant results that utilize the most general possible notion of a perversity asa function

Np W fcomponents of singular strata of a stratified pseudomanifoldg ! Z:

In this setting, one usually loses topological invariance of intersection homol-ogy (though this should be seen not as a loss but as an opportunity to studystratification data), but duality results remain, at least if one chooses the rightgeneralizations of intersection homology. Complicating this choice is the factthat there are a variety of approaches to intersection homology to begin with,even using Goresky and MacPherson’s perversities. These include (at the least)the original simplicial chain definition [32]; Goresky and MacPherson’s Delignesheaves [33; 6]; King’s singular chain intersection homology [32]; Cheeger’sL2

cohomology andL2 Hodge theory [16]; perverse differential forms on Thom–Mather stratified spaces (and, later, on unfoldable spaces [7]), first publishedby Brylinski [8] but attributed to Goresky and MacPherson; and the theory ofperverse sheaves [4]. Work to find the “correct” versions of these theories whengeneral perversities are allowed has been performed by the author, using strati-fied coefficients for simplicial and singular intersection chains [26]; by Saralegi,using “relative” intersection homology and perverse differential forms in [54];and by the author, generalizing the Deligne sheaf in [22]. Special cases of non-Goresky–MacPherson perversities in theL2 Hodge theory setting have also beenconsidered by Hausel, Hunsicker, and Mazzeo [37]; Hunsicker and Mazzeo [39];and Hunsicker [38]. And arbitrary perversities have been available from the startin the theory of perverse sheaves!

This paper is intended to serve as something of a guidebook tothe differentnotions of perversities and as an introduction to some new and exciting work inthis area. Each stage of development of the idea of perversities was accompaniedby a flurry of re-examinings of what it means to have an intersection homologytheory and what spaces such a theory can handle as input, and each such re-examining had to happen within one or more of the contexts listed above. In

INTERSECTION HOMOLOGY WITH GENERAL PERVERSITY FUNCTIONS 179

many cases, the outcome of this re-examination led to a modification or ex-pansion of the basic definitions. This has resulted in a, quite justified, paradeof papers consumed with working through all the technical details. However,technicalities often have the unintended effect of obscuring the few key mainideas. Our goal then is to present these key ideas and their consequences inan expository fashion, referring the reader to the relevantpapers for furthertechnical developments and results. We hope that such a survey will providesomething of an introduction to and overview of the recent and ongoing workof the author, but we also hope to provide a readable (and hopefully accurate!)historical account of this particular chain of ideas and an overview of the workof the many researchers who have contributed to it. We additionally hope thatsuch an overview might constitute a suitable introduction for those wishing tolearn about the basics of intersection homology and as preparation for thosewishing to pursue the many intriguing new applications thatgeneral perversitiesbring to the theory.

This exposition is not meant to provide a comprehensive historical accountbut merely to cover one particular line of development. We will focus primarilyon the approaches to intersection homology by simplicial and singular chainsand by sheaf theory. We will touch only tangentially upon perverse differentialforms when we consider Saralegi’s work in Section 10; we advise the reader toconsult [54] for the state of the art, as well as references toprior work, in thisarea. Also, we will not discussL2-cohomology. This is a very active field ofresearch, as is well-demonstrated elsewhere in this volume[30], but the study ofL2-cohomology andL2 Hodge theories that yield intersection homology withgeneral perversities remains under development. The reader should consult thepapers cited above for the work that has been done so far. We will briefly discussperverse sheaves in Section 8.2, but the reader should consult [4] or any of thevariety of fine surveys on perverse sheaves that have appeared since for moredetails.

We will not go into many of the myriad results and applications of inter-section homology theory, especially those beyond topologyproper in analysis,algebraic geometry, and representation theory. For broader references on inter-section homology, the reader might start with [6; 42; 2]. These are also excellentsources for the material we will be assuming regarding sheaftheory and derivedcategories and functors.

We proceed roughly in historical order as follows: Section 2provides theoriginal Goresky–MacPherson definitions of PL pseudomanifolds and PL chainintersection homology. We also begin to look closely at the cone formula forintersection homology, which will have an important role toplay throughout. InSection 3, we discuss the reasons for the original Goresky–MacPherson condi-

180 GREG FRIEDMAN

tions on perversities and examine some consequences, and weintroduce King’ssingular intersection chains. In Section 4, we turn to the sheaf-theoretic defi-nition of intersection homology and introduce the Deligne sheaf. We discussthe intersection homology version of Poincare duality, then we look at ourfirst example of an intersection homology result that utilizes a non-Goresky–MacPherson perversity, the Cappell–Shaneson superduality theorem.

In Section 5, we discusssubperversitiesandsuperperversities. Here we firstobserve the schism that occurs between chain-theoretic andsheaf-theoretic in-tersection homology when perversities do not satisfy the Goresky–MacPhersonconditions. Section 6 introducesstratified coefficients, which were developedby the author in order to correct the chain version of intersection homology forit to conform with the Deligne sheaf version.

In Section 7, we discuss the further evolution of the chain theory to the mostgeneral possible perversities and the ensuing results and applications. Section 8contains the further generalization of the Deligne sheaf togeneral perversities,as well as a brief discussion of perverse sheaves and how general perversityintersection homology arises in that setting. Some indications of recent workand work-in-progress with these general perversities is provided in Section 9.

Finally, Sections 10 and 11 discuss some alternative approaches to intersec-tion homology with general perversities. In Section 10, we discuss Saralegi’s“relative intersection chains”, which are equivalent to the author’s stratified co-efficients when both are defined. In Section 11, we present thework of Habeggerand Saper from [35]. This work encompasses another option tocorrecting theschism presented in Section 5 by providing a sheaf theory that agrees with King’ssingular chains, rather than the other way around; however,the Habegger–Sapertheory remains rather restrictive with respect to acceptable perversities.

2. The original definition of intersection homology

We begin by recalling the original definition of intersection homology asgiven by Goresky and MacPherson in [32]. We must start with the spaces thatintersection homology is intended to study.

2.1. Piecewise linear stratified pseudomanifolds.The spaces considered byGoresky and MacPherson in [32] werepiecewise linear (PL) stratified pseudo-manifolds. An n-dimensional PL stratified pseudomanifoldX is a piecewiselinear space (meaning it is endowed with a compatible familyof triangulations)that also possesses a filtration by closed PL subspaces (the stratification)

X DX n �X n�2 �X n�3 � � � � �X 1 �X 0 �X �1 D?

satisfying the following properties:


(a) X �X n�2 is dense inX ,(b) for eachk � 2, X n�k �X n�k�1 is either empty or is ann�k dimensional

PL manifold,(c) if x 2X n�k �X n�k�1, thenx has adistinguished neighborhoodN that is

PL homeomorphic toRn�k � cL, wherecL is the open cone on a compactk�1 dimensional stratified PL pseudomanifoldL. Also, the stratification ofL must be compatible with the stratification ofX .

A PL stratified pseudomanifoldX is oriented (or orientable) ifX �X n�2

has the same property.

A few aspects of this definition deserve comment. Firstly, the definition is induc-tive: to define ann-dimensional PL stratified pseudomanifold, we must alreadyknow what ak � 1 dimensional PL stratified pseudomanifold is fork � 1 < n.The base case occurs fornD 0; a 0-pseudomanifold is a discrete set of points.Secondly, there is a gap fromn to n� 2 in the filtration indices. This is more-or-less intended to avoid issues of pseudomanifolds with boundary, althoughthere are now established ways of dealing with these issues that we will returnto below in Section 5.

The setsX i are calledskeleta, and we can verify from condition (b) thateach has dimension� i as a PL complex. The setsXi WD X i � X i�1 aretraditionally calledstrata, though it will be more useful for us to use this term forthe connected components ofXi , and we will favor this latter usage rather thanspeaking of “stratum components.”1 The strata ofX n�X n�2 are calledregularstrata, and the other strata are calledsingular strata. The spaceL is called thelink of x or of the stratum containingx. For a PL stratified pseudomanifoldLis uniquely determined up to PL homeomorphism by the stratumcontainingx.The conecL obtains a natural stratification from that ofL: .cL/0 is the conepoint and fori > 0, .cL/i DLi�1 � .0; 1/� cL, where we think ofcL as

L� Œ0; 1/

.x; 0/� .y; 0/:

The compatibility condition of item (c) of the definition means that the PL home-omorphism should takeX i \N to R

n�k � .cL/i�.n�k/.Roughly, the definition tells us the following. Ann-dimensional PL stratified

pseudomanifoldX is mostly then-manifold X �X n�2, which is dense inX .(In much of the literature,X n�2 is also referred to as , the singular locus ofX .) The rest ofX is made up of manifolds of various dimensions, and thesemust fit together nicely, in the sense that each point in each stratum should have

1It is perhaps worth noting here that the notation we employ throughout mostly willbe consistent with the author’s own work, though not necessarily with all historicalsources.

182 GREG FRIEDMAN

a neighborhood that is a trivial fiber bundle, whose fibers arecones on lower-dimensional stratified spaces.

We should note that examples of such spaces are copious. Any complex an-alytic or algebraic variety can be given such a structure (see [6, Section IV]), ascan certain quotient spaces of manifolds by group actions. PL pseudomanifoldsoccur classically as spaces that can be obtained from a pile of n-simplices bygluing in such a way that eachn� 1 face of ann-simplex is glued to exactlyonen� 1 face of one othern-simplex. (Another classical condition is that weshould be able to move from any simplex to any other, passing only throughinteriors ofn�1 faces. This translates to say thatX �X n�2 is path connected,but we will not concern ourselves with this condition.) Other simple examplesarise by taking open cones on manifolds (naturally, given the definition), bysuspending manifolds (or by repeated suspensions), by gluing manifolds andpseudomanifolds together in allowable ways, etc. One can construct many use-ful examples by such procedures as “start with this manifold, suspend it, crossthat with a circle, suspend again, . . . ” For more detailed examples, the readermight consult [6; 2; 44].

More general notions of stratified spaces have co-evolved with the variousapproaches to intersection homology, mostly by dropping orweakening require-ments. We shall attempt to indicate this evolution as we progress.

2.2. Perversities. Besides the spaces on which one is to define intersectionhomology, the other input is the perversity parameter. In the original Goresky–MacPherson definition, a perversityNp is a function from the integers� 2 to thenon-negative integers satisfying the following properties:

(a) Np.2/D 0.(b) Np.k/� Np.kC 1/� Np.k/C 1.

These conditions say that a perversity is something like a sub-step function.It starts at0, and then each time the input increases by one, the output eitherstays the same or increases by one. Some of the most commonly used perver-sities include the zero perversityN0.k/ D 0, the top perversityNt.k/ D k � 2,the lower-middle perversityNm.k/ D

�

k�22

˘

, and the upper middle perversityNn.k/D

�

k�12

˘

.The idea of the perversity is that the input numberk represents the codimen-

sion of a stratumXn�k D X n�k �X n�k�1 of an n-dimensional PL stratifiedpseudomanifold, while the output will control the extent towhich the PL chainsin our homology computations will be permitted to interact with these strata.

The reason for the arcane restrictions onNp will be made clear below in Sec-tion 3. We will call any perversity satisfying conditions (a) and (b) aGoresky–MacPherson perversity, or aGM perversity.


2.3. Intersection homology. At last, we are ready to discuss intersection ho-mology.

Let X be ann-dimensional PL stratified pseudomanifold, and letC T�.X /

denote the simplicial chain complex ofX with respect to the triangulationT .The PL chain complexC�.X / is defined to be lim

�T C T�.X /, where the limit is

taken with respect to the directed set of compatible triangulations. This PL chaincomplex is utilized by Goresky and MacPherson in [32] (see also [6]), and it isuseful in a variety of other contexts (see [46], for instance). However, it turns outthat this is somewhat technical overkill for the basic definition of intersectionhomology, as what follows can also be performed inC T

�.X /, assumingT is

sufficiently refined with respect to the the stratification ofX (for example, pickanyT , take two barycentric subdivisions, and you’re set to go — see [45]).

We now define the perversityNp intersection chain complexI NpC�.X / �

C�.X /. We say that a PLj -simplex� is Np-allowableprovided

dim.� \Xn�k/� j � kC Np.k/

for all k � 2. We say that a PLi-chain � 2 Ci.X / is Np-allowable if each i-simplex occurring with nonzero coefficient in� is Np-allowable and if eachi �1

simplex occurring with nonzero coefficient in@� is Np-allowable. Notice that thesimplices in� must satisfy the simplex allowability condition withj D i whilethe simplices of@� must satisfy the condition withj D i � 1.

ThenI NpC�.X / is defined to be the complex of allowable chains. It followsimmediately from the definition that this is indeed a chain complex. The inter-section homology groups areI NpH�.X /DH�.I

NpC�.X //.Some remarks are in order.

REMARK 2.1. The allowability condition at first seems rather mysterious. How-ever, the condition dim.� \Xn�k/� j �k would be precisely the requirementthat� andXn�k intersect in general position ifXn�k were a submanifold ofX .Thus introducing a perversity can be seen as allowing deviation from generalposition to a degree determined by the perversity. This seems to be the originof the nomenclature.

REMARK 2.2. It is a key observation that if� is ani-chain, then it is not everyi � 1 face of everyi-simplex of� that must be checked for its allowability, butonly those that survive in@�. Boundary pieces that cancel out do not need to bechecked for allowability. This seemingly minor point accounts for many subtlephenomena, including the next remark.

REMARK 2.3. Intersection homology with coefficientsI NpH�.X IG/ can bedefined readily enough beginning withC�.X IG/ instead ofC�.X /. However,I NpC�.X IG/ is generallyNOT the same asI NpC�.X /˝G. This is precisely due

184 GREG FRIEDMAN

to the boundary cancellation behavior: extra boundary cancellation in chainsmay occur whenG is a group with torsion, leading to allowable chains inI NpC�.X IG/ that do not come from anyG-linear combinations of allowablechains inI NpC�.X IZ/. For more details on this issue, including many examples,the reader might consult [29].

REMARK 2.4. In [32], Goresky and MacPherson stated the allowability condi-tion in terms of skeleta, not strata. In other words, they define aj -simplex tobe allowable if

dim.� \X n�k/� j � kC Np.k/

for all k � 2. However, it is not difficult to check that the two conditionsareequivalent for the perversities we are presently considering. When we move onto more general perversities, below, it becomes necessary to state the conditionin terms of strata rather than in terms of skeleta.

2.4. Cones. It turns out that understanding cones plays a crucial role inal-most all else in intersection homology theory, which perhaps should not be toosurprising, as pseudomanifolds are all locally products ofcones with euclideanspace. Most of the deepest proofs concerning intersection homology can bereduced in some way to what happens in these distinguished neighborhoods.The euclidean part turns out not to cause too much trouble, but cones possessinteresting and important behavior.

So letL be a compactk � 1 dimensional PL stratified pseudomanifold, andlet cL be the open cone onL. Checking allowability of aj -simplex� withrespect to the cone vertexfvg D .cL/0 is a simple matter, since the dimensionof � \ fvg can be at most0. Thus� can allowably intersectv if and only if0� j�kC Np.k/, i.e., if j �k� Np.k/. Now, suppose� is an allowablei-cycle inL. We can form the chainNc� 2 I NpCiC1.cL/ by taking the cone on each simplexin the chain (by extending each simplex linearly to the cone point). We cancheck using the above computation (and a little more work that we’ll suppress)that Nc� is allowable ifi C 1 � k � Np.k/, and thus� D @ Nc� is a boundary; see[6, Chapters I and II]. Similar, though slightly more complicated, computationsshow that any allowable cycle incL is a boundary. ThusI NpHi.cL/ D 0 ifi � k �1� Np.k/. On the other hand, ifi < k�1� Np.k/, then noi-chain� canintersectv nor can any chain of which it might be a boundary. Thus� is left toits own devices incL�v, i.e.,I NpHi.cL/D I NpHi.cL�v/Š I NpHi.L� .0; 1//.It turns out that intersection homology satisfies the Kunneth theorem when onefactor is euclidean space and we take the obvious product stratification (see [6,Chapter I]), or alternatively we can use the invariance of intersection homologyunder stratum-preserving homotopy equivalences (see [23]), and so in this rangeI NpHi.cL/Š I NpHi.L/.


Altogether then, we have

I NpHi.cLk�1/Š

�

0 if i � k � 1� Np.k/,I NpHi.L/ if i < k � 1� Np.k/.

(2-1)

We will return to this formula many times.

3. Goresky–MacPherson perversities

The reasons for the original Goresky–MacPherson conditions on perversi-ties, as enumerated in Section 2.2, are far from obvious. Ultimately, they comedown to the two initially most important properties of intersection homology:its topological invariance and its Poincare duality.

The topological invariance property of traditional intersection homology saysthat when Np is a Goresky–MacPherson perversity andX is a stratified pseu-domanifold (PL or topological, as we’ll get to soon) thenI NpH�.X / dependsonly onX and not on the choice of stratification (among those allowed by thedefinition). This is somewhat surprising considering how the intersection chaincomplex depends on the strata.

The desire forI NpH�.X / to be a topological invariant leads fairly quickly tothe condition that we should not allowNp.k/ to be negative. This will be moreevident once we get to the sheaf-theoretic formulation of intersection homology,but for now, consider the cone formula (2-1) forcLk�1, and supposeNp.k/ < 0.Then we can check that no allowable PL chain may intersectv. Thus we see thatthe intersection homology ofcL is the same as if we removed the cone pointaltogether. A little more work (see [22, Corollary 2.5]) leads more generally tothe conclusion that ifNp.k/ < 0, thenI NpH�.X /Š I NpH�.X �Xk/. This wouldviolate the topological invariance since, for example, topological invariance tellsus that if M n is a manifold thenI NpH�.M / Š H�.M /, no matter how westratify it2. But if we now allow, say, a locally-flat PL submanifoldN n�k andstratify byM � N , then if Np.k/ < 0 we would haveH�.M / Š I NpH�.M /Š

I NpH�.M �N / Š H�.M �N /. This presents a clear violation of topologicalinvariance.

The second Goresky–MacPherson condition, thatNp.k/� Np.kC1/� Np.k/C1,also derives from topological invariance considerations.The following exampleis provided by King [41, p. 155]. We first note that, lettingSX denote thesuspension ofX , we havecSX ŠR� cX (ignoring the stratifications). This isnot hard to see topologically (recall thatcX is theopencone onX ). But now ifwe assumeX is k � 1 dimensional and that we take the obvious stratifications

2Note that one choice of stratification is the trivial one containing a single regularstratum, in which case it is clear from the definition thatI NpH�.M /ŠH�.M /.

186 GREG FRIEDMAN

of R � cX (assuming some initial stratification onX ), then

I NpHi.R � cX /Š

�

0 if i � k � 1� Np.k/,I NpHi.X / if i < k � 1� Np.k/.

(3-1)

This follows from the cone formula (2-1) together with the intersection ho-mology Kunneth theorem, for which one term is unstratified [41] (or stratum-preserving homotopy equivalence [23]).

But now it also follows by an easy argument, using (2-1) and the Mayer–Vietoris sequence, that

I NpHi.SX /Š

8

<

:

I NpHi�1.X / if i > k � 1� Np.k/,0 if i D k � 1� Np.k/,I NpHi.X / if i < k � 1� Np.k/,

(3-2)

and, sinceSX has dimensionk,

I NpHi.cSX /Š

�

0 if i � k � Np.kC 1/,I NpHi.SX / if i < k � Np.kC 1/.

(3-3)

So,I NpHi.R � cX / is 0 for i � k � 1� Np.k/, while I NpHi.cSX / must be0 fori �k� Np.kC1/ and also foriDk�1� Np.k/ even ifk�1� Np.k/<k� Np.kC1/.Also, it is not hard to come up with examples in which the termsthat are notforced to be zero are, in fact, nonzero. Ifk � 1� Np.k/ � k � Np.k C 1/ (i.e.,1C Np.k/ � Np.k C 1/), so that the special casei D k � 1 � Np.k/ is alreadyin the zero range forI NpH�.cSX /, then topological invariance would requirek � 1 � Np.k/ D k � Np.k C 1/, i.e., Np.k C 1/ D Np.k/ C 1. So if we wanttopological invariance,Np.kC 1/ cannot be greater thanNp.k/C 1.

On the other hand, ifk�1� Np.k/<k� Np.kC1/, the0 atI NpHk�1� Np.k/.cSX /

forced by the suspension formula drops below the truncationdimension cutoff atk� Np.kC1/ that arises from the cone formula. Ifk�1� Np.k/Dk�1� Np.kC1/

(i.e., Np.k/D Np.kC 1/), no contradiction occurs. But if

k � 1� Np.k/ < k � 1� Np.kC 1/

(i.e., Np.kC 1/ < Np.k/), thenI NpHk�1� Np.kC1/.cSX / could be nonzero, whichmeans, via the formula forI NpH�.R�cX /, that we must havek�1� Np.kC1/<

k � 1� Np.k/ (i.e., Np.kC 1/ > Np.k/), yielding a contradiction.Hence the only viable possibilities for topological invariance areNp.kC1/D

Np.k/ or Np.kC 1/D Np.k/C 1.It turns out that both possibilities work out. Goresky and MacPherson [33]

showed using sheaf theory that any perversity satisfying the two Goresky–Mac-Pherson conditions yields a topologically invariant intersection homology the-ory. King [41] later gave a non-sheaf proof that holds even when Np.2/ > 0.


Why, then, did Goresky and MacPherson limit consideration to perversitiesfor which Np.2/ D 0? For one thing, they were primarily concerned with thePoincare duality theorem for intersection homology, which states that if X isa compact orientedn-dimensional PL stratified pseudomanifold, then there is anondegenerate pairing

I NpHi.X IQ/˝ I NqHn�i.X IQ/!Q

if Np and Nq satisfy the Goresky–MacPherson conditionsand Np C Nq D Nt , or, inother words,Np.k/C Nq.k/D k�2. If we were to try to allow Np.2/ > 0, then wewould have to haveNq.2/ < 0, and we have already seen that this causes troublewith topological invariance. So if we want both duality and invariance, we musthave Np.2/ D Nq.2/ D 0. Without this condition we might possibly have one orthe other, but not both. In fact, King’s invariance results for Np.2/ > 0 impliesthat duality cannot hold in general when we pair a perversitywith Np.2/> 0 withone with Nq.2/ < 0, at least not without modifying the definition of intersectionhomology, which we do below.

But there is another interesting reason that Goresky and MacPherson did notobtain King’s invariance result forNp.2/ > 0. When intersection homology wasfirst introduced in [32], Goresky and MacPherson were unableinitially to provetopological invariance. They eventually succeeded by reformulating intersectionhomology in terms of sheaf theory. But, as it turns out, whenNp.2/ ¤ 0 theoriginal sheaf theory version of intersection homology does not agree with thechain version of intersection homology we have been discussing and for whichKing proved topological invariance. Furthermore, the sheaf version is not atopological invariant whenNp.2/ > 0 (some examples can be found in [24]).Due to the powerful tools that sheaf theory brings to intersection homology,the sheaf theoretic point of view has largely overshadowed the chain theory.However, this discrepancy between sheaf theory and chain theory for non-GMperversities turns out to be very interesting in its own right, as we shall see.

3.1. Some consequences of the Goresky–MacPherson conditions. The Go-resky–MacPherson perversity conditions have a variety of interesting conse-quences beyond turning out to be the right conditions to yield both topologicalinvariance and Poincare duality.

Recall that the allowability condition for ani-simplex� is that dim.�\Xk/�

i � kC Np.k/. The GM perversity conditions ensure thatNp.k/ � k � 2, and sofor any perversity we must havei � k C Np.k/ � i � 2. Thus noi-simplex inan allowable chain can intersect any singular stratum in theinteriors of itsi orits i �1 faces. One simple consequence of this is that no0- or 1-simplices mayintersectX n�2, and soI NpH0.X /ŠH0.X �X n�2/.

188 GREG FRIEDMAN

Another consequence is the following fantastic idea, also due to Goreskyand MacPherson. Suppose we have a local coefficient system ofgroups (i.e.,a locally constant sheaf) defined onX �X n�2, even perhaps one that cannotbe extended to all ofX . If one looks back at early treatments of homologywith local coefficient systems, for example in Steenrod [56], it is sufficient toassign a coefficient group to each simplex of a triangulation(we can think ofthe group as being located at the barycenter of the simplex) and then to assignto each boundary face map a homomorphism between the group onthe simplexand the group on the boundary face. This turns out to be sufficient to definehomology with coefficients — what happens on lower dimensional faces doesnot matter (roughly, everything on lower faces cancels out because we still have@2 D 0). Since the intersectioni-chains with the GM perversities have thebarycenters of their simplices and of their topi � 1 faces outside ofX n�2, alocal coefficient systemG on X n�2 is sufficient to define the intersection chaincomplexI NpC�.X IG/ and the resulting homology groups. For more details onthis construction, see, [26], for example.

Of course now the stratification does matter to some extent since it determineswhere the coefficient system is defined. However, see [6, Section V.4] for adiscussion of stratifications adapted to a given coefficientsystem defined on anopen dense set ofX of codimension� 2.

One powerful application of this local coefficient version of intersection ho-mology occurs in [12], in which Cappell and Shaneson study singular knots byconsidering the knots in their ambient spaces as stratified spaces. They employa local coefficient system that wraps around the knot to mimicthe coveringspace arguments of classical knot theory. This work also contains one of thefirst useful applications of intersection homology with non-GM perversities. Inorder to explain this work, though, we first need to discuss the sheaf formulationof intersection homology, which we pick up in Section 4.

3.2. Singular chain intersection homology. Before moving on to discussthe sheaf-theoretic formulation of intersection homology, we jump ahead in thechronology a bit to King’s introduction of singular chain intersection homologyin [41]. As one would expect, singular chains are a bit more flexible than PLchains (pun somewhat intended), and the singular intersection chain complexcan be defined on any filtered spaceX �X n�1 �X n�2 � � � � , with no furtherrestrictions. In fact, the “dimension” indices of the skeleta X k need no longerhave a geometric meaning. These spaces include both PL stratified pseudoman-ifolds andtopological stratified pseudomanifolds, the definition of which is thesame as of PL pseudomanifolds but with all requirements of piecewise linearitydropped. We also extend the previous definition now to allow an n�1 skeleton,and we must extend perversities accordingly to be functionsNp WZ�1!Z. King


definesloose perversities, which are arbitrary functions of this type. We willreturn to these more general perversities in greater detailas we go on.

To define the singular intersection chain complex, which we will denoteI NpS�.X /, we can no longer use dimension of intersection as a criterion (es-pecially if the index of a skeleton no longer has a dimensional meaning). In-stead, the natural generalization of the allowability condition is that a singulari-simplex� W�i !X is allowable if

��1.Xn�k/� fi � kC Np.k/ skeleton of�ig:

Once allowability has been defined for simplices, allowability of chains is de-fined as in the PL case, and we obtain the chain complexI NpS�.X / and thehomology groupsI NpH�.X /.

If X is a PL stratified pseudomanifold, the notationI NpH�.X / for singularchain intersection homology causes no confusion; as King observes, the PL andsingular intersection homology theories agree on such spaces. Also as for PLchains, and by essentially the same arguments, ifX has no codimension onestratum andNp is a GM perversity, singular intersection homology can takelocalcoefficients onX �X n�2.

From here on, when we refer to chain-theoretic intersectionhomology, wewill mean both the singular version (in any context) and the PL version (on PLspaces).

4. Sheaf-theoretic intersection homology

Although intersection homology was developed originally utilizing PL chaincomplexes, this approach was soon largely supplanted by thetechniques of sheaftheory. Sheaf theory was brought to bear by Goresky and MacPherson in [33],originally as a means to demonstrate the topological invariance (stratificationindependence) of intersection homology with GM perversities; this was beforeKing’s proof of this fact using singular chains. However, itquickly becameevident that sheaf theory brought many powerful tools alongwith it, including aVerdier duality approach to the Poincare duality problem on pseudomanifolds.Furthermore, the sheaf theory was able to accommodate topological pseudo-manifolds. This sheaf-theoretic perspective has largely dominated intersectionhomology theory ever since.

The Deligne sheaf. We recall that ifX n is a stratified topological pseudo-manifold3, then a primary object of interest is the so-calledDeligne sheaf. Fornotation, we letUk DX �X n�k for k � 2, and we letik WUkŒUkC1 denote

3For the moment, we again make the historical assumption thatthere are no codi-mension one strata.

190 GREG FRIEDMAN

the inclusion. Suppose thatNp is a GM perversity. We have seen that intersectionhomology should allow a local system of coefficients defined only onX�X n�2;let G be such a local system. The Deligne sheaf complexP� (or, more preciselyP�

Np;G) is defined by an inductive process. It is4

P� D �� Np.n/Rin� : : : �� Np.2/Ri2�.G˝O/;

whereO is the orientation sheaf onX �X n�2, Rik� is the right derived functorof the pushforward functorik�, and��m is the sheaf complex truncation functorthat takes the sheaf complexS� to ��mS� defined by

.��mS�/i D

(

0 if i >m,ker.di/ if i Dm,Si if i <m.

Heredi is the differential of the sheaf complex. Recall that��mS� is quasi-isomorphic toS� in degrees�m and is quasi-isomorphic to0 in higher degrees.

REMARK 4.1. Actually, the orientation sheafO is not usually included here aspart of the definition ofP�, or it would be only if we were discussingP�

Np;G˝O.

However, it seems best to include this here so as to eliminatehaving to continu-ally mess with orientation sheaves when discussing the equivalence of sheaf andchain theoretic intersection homology, which, without this convention, wouldread thatH�.X IP�

Np;G˝O/ Š I NpHn��.X IG/; see below. PuttingO into the

definition ofP� as we have done here allows us to leave this nuisance tacit inwhat follows.

The connection between the Deligne sheaf complex (also called simply the“Deligne sheaf”) and intersection homology is that it can beshown that, on ann-dimensional PL pseudomanifold,P� is quasi-isomorphic to the sheafU !

I NpC 1n��.U IG/. Here the1 indicates that we are now working with Borel–

Moore PL chain complexes, in which chains may contain an infinite number ofsimplices with nonzero coefficients, so long as the collection of such simplicesin any chain is locally-finite. This is by contrast to the PL chain complex dis-cussed above for which each chain can contain only finitely many simplices withnonzero coefficient. This sheaf of intersection chains is also soft, and it followsvia sheaf theory that the hypercohomology of the Deligne sheaf is isomorphicto the Borel–Moore intersection homology

H�.X IP�/Š I NpH 1

n��.X IG/:

4There are several other indexing conventions. For example,it is common to shiftthis complex so that the coefficientsG live in degree�n and the truncations become�� Np.k/�n. There are other conventions that make the cohomologicallynontrivial degreesof the complex symmetric about0 whenn is even. We will stick with the convention thatG lives in degree0 throughout. For details on other conventions, see [33], forexample.


It is also possible to recover the intersection homology we introduced initiallyby using compact supports:

H�

c .X IP�/Š I NpH c

n��.X IG/:

Now that we have introduced Borel–Moore chains, we will use “c” to indicatethe more familiar compact (finite number of simplices) supports. If the resultswe discuss hold in both contexts (in particular ifX is compact) we will forgoeither decoration. More background and details on all of this can be found in[33; 6].

It was shown later, in [26], that a similar connection existsbetween theDeligne sheaf and singular chain intersection homology ontopologicalpseudo-manifolds. Continuing to assume GM perversities, one can also define a sheafvia the sheafification of the presheaf ofsingular chains5

U ! I NpSn��.X;X � NU IG/:

This sheaf turns out to be homotopically fine, and it is again quasi-isomorphicto the Deligne sheaf. Thus, once again, we have

H�

c .X IP�/Š I NpH c

n��.X IG/ and H�.X IP�/Š I NpH 1

n��.X IG/;

which is the homology of the chain complexI NpS1�.X IG/ consisting of chains

that can involve an infinite, though locally-finite, number of simplices withnonzero coefficient.

The Goresky–MacPherson proof of topological invariance follows by show-ing that the Deligne sheaf is uniquely defined up to quasi-isomorphism via aset of axioms that do not depend on the stratification of the space. This proofis given in [33]. However, we would here like to focus attention on what theDeligne sheaf accomplishes locally, particularly in mind of the maxim that asheaf theory (and sheaf cohomology) is a machine for assembling local in-formation into global. So let’s look at the local cohomology(i.e., the stalkcohomology) of the sheafP� at x 2 Xn�k . This isH�.P�/x D H �.P�

x/ Š

lim�x2U H�.U IP�/ Š lim

�x2U I NpH 1n��

.U IG/, and we may assume that thelimit is taken over the cofinal system of distinguished neighborhoodsN Š

Rn�k � cLk�1 containingx. It is not hard to see thatP� at x 2Xn�k depends

only on the stages of the iterative Deligne construction up through�� Np.k/Rik�

(at least so long as we assume thatNp is nondecreasing6, as it will be for aGM perversity). Then it follows immediately from the definition of � thatH�.P�/x D 0 for � > Np.k/. On the other hand, the pushforward construction,

5SinceX is locally compact, we may use eitherc or1 to obtain the same sheaf.6If Np ever decreases, say atk, then the truncation�� Np.k/ might kill local cohomology

in other strata of lower codimension.

192 GREG FRIEDMAN

together with a Kunneth computation and an appropriate induction step (see [6,Theorem V.2.5]), shows that for� � Np.k/ we have

H�.P�/x ŠH

�.N �N \X n�k IP�/

ŠH�.Rn�k � .cL� v/IP�/

ŠH�.Rn�kC1�LIP�/

ŠH�.LIP�jL/:

It can also be shown thatP�jL is quasi-isomorphic to the Deligne sheaf onL,soH�.LIP�jL/Š I NpHk�1��.L/.

For future reference, we record the formula

Hi.P�/x Š

�

0 if i > Np.k/,Hi.LIP�/ if i � Np.k/,

(4-1)

for x 2 Xn�k andL the link of x. Once one accounts for the shift in indexingbetween intersection homology and Deligne sheaf hypercohomology and forthe fact that we are now working with Borel–Moore chains, these computationswork out to be equivalent to the cone formula (2-1). In fact,

H �.P�

x/Š I NpH 1

n��.Rn�k � cLIG/

Š I NpH 1

k��.cLIG/ (by the Kunneth theorem)

Š I NpHk��.cL;L� .0; 1/IG/;

and the cone formula (2-1) translates directly, via the longexact sequence ofthe pair.cL;L � .0; 1//, to this being0 for � > Np.k/ andI NpHk�1��.LIG/

otherwise.So the Deligne sheaf recovers the local cone formula, and onewould be hard

pressed to find a more direct or natural way to “sheafify” the local cone conditionthan the Deligne sheaf construction. This reinforces our notion that the coneformula is really at the heart of intersection homology. In fact, the axiomaticcharacterization of the Deligne sheaf alluded to above is strongly based uponthe sheaf version of the cone formula. There are several equivalent sets of char-acterizing axioms. The first,AX 1 Np;G , is satisfied by a sheaf complexS� if

(a) S� is bounded andS� D 0 for i < 0,(b) S�jX �X n�2 Š G˝O,7

(c) for x 2Xn�k , H i.S�x/D 0 if i > Np.k/, and

(d) for each inclusionik W Uk ! UkC1, the “attaching map” k given my thecomposition of natural morphismsS�jUkC1

! ik�i�

kS� ! Rik�i�

kS� is a

quasi-isomorphism in degrees� Np.k/.

7See Remark 4.1 on page 190.


These axioms should technically be thought of as applying inthe derived cate-gory of sheaves onX , in which case all equalities and isomorphisms should bethought of as quasi-isomorphisms of sheaf complexes. The first axiom acts assomething of a normalization and ensures thatS� lives in the bounded derivedcategory. The second axiom fixes the coefficients onX �X n�2. The third andfourth axioms are equivalent to the cone formula (4-1); see [6, Sections V.1 andV.2]. In fact, it is again not difficult to see that the Delignesheaf constructionis designed precisely to satisfy these axioms. It turns out that these axiomscompletely characterize a sheaf up to quasi-isomorphism (see [6, Section V.2]),and in fact it is by showing that the sheafification ofU ! I NpS�.X;X � NU IG/

satisfies these axioms that one makes the connection betweenthe sheaf of sin-gular intersection chains and the Deligne sheaf.

Goresky and MacPherson [33] proved the stratification independence of in-tersection homology by showing that the axiomsAX 1 are equivalent to othersets of axioms, including one that does not depend on the stratification of X .See [6; 33] for more details.

4.1. Duality. It would take us too far afield to engage in a thorough discussionof how sheaf theory and, in particular, Verdier duality leadto proofs of theintersection homology version of Poincare duality. However, we sketch someof the main ideas, highlighting the role that the perversityfunctions play in thetheory. For complete accounts, we refer the reader to the excellent expositorysources [6; 2].

The key to sheaf-theoretic duality is the Verdier dualizingfunctionD. Veryroughly,D functions as a fancy sheaf-theoretic version of the functorHom.�;R/.In fact,D takes a sheaf complexS� to a sheaf complexHom�.S�;D�

X/, where

D�

Xis the Verdier dualizing sheaf on the spaceX . In reasonable situations,

the dualizing sheafD�

Xis quasi-isomorphic (after reindexing) to the sheaf of

singular chains onX ; see [6, Section V.7.2.]. For us, the most important propertyof the functorD is that it satisfies a version of the universal coefficient theorem.In particular, ifS� is a sheaf complex over the Dedekind domainR, then forany openU �X ,

Hi.U IDS

�/Š Hom.H�ic .U IS

�/IR/˚Ext.H�iC1c .U IS�/IR/:

The key, now, to proving a duality statement in intersectionhomology isto show that ifX is orientable over a ground fieldF and Np and Nq are dualperversities, meaningNp.k/C Nq.k/D k�2 for all k � 2, thenDP�

Np Œ�n� is quasi-

isomorphic toP�

Nq . HereŒ�n� is the degree shift by�n degrees, i.e.,.S�Œ�n�/iD

Si�n, and this shift is applied toDP� (it is not a shiftedP� being dualized).It then follows from the universal coefficient theorem with field coefficientsF

194 GREG FRIEDMAN

that

I NqH 1

n�i.X IF /Š Hom.I NpH ci .X IF /;F /;

which is intersection homology Poincare duality for pseudomanifolds.To show thatDP�

Np Œ�n� is quasi-isomorphic toP�

Nq , it suffices to show thatDP�

Np Œ�n� satisfies the axiomsAX 1 Nq. Again, we will not go into full detail, butwe remark the following main ideas, referring the reader to the axiomsAX 1

outlined above:

(a) OnX �X n�2, DP�

Np Œ�n� restricts to the dual of the coefficient system

P�

NpjX �X n�2 ;

which is again a local coefficient system. IfX is orientable and the coefficientsystem is trivial, then so is its dual.

(b) Recall that the third and fourth axioms for the Deligne sheaf concern whathappens at a pointx in the stratumXn�k . To computeH �.S�

x/, we maycompute lim

�x2U H�.U IS�/. In particular, if we let eachU be a distin-guished neighborhoodU ŠR

n�k�cL of x and apply the universal coefficienttheorem, we obtain

H i.U IDP�

Np Œ�n�x/Š lim�x2U H

i.DP�

Np Œ�n�/

Š lim�x2U H

i�n.U IDP�

Np/

Š lim�x2U Hom.Hn�i

c .U IP�

Np/;F /

Š lim�x2U Hom.I NpH c

i .Rn�k � cLIF /;F /

Š lim�x2U Hom.I NpH c

i .cLIF /;F /:

(4-2)

The last equality is from the Kunneth theorem with compact supports. Fromthe cone formula, we know that this will vanish ifi � k � 1� Np.k/, i.e., ifi > k � 2� Np.k/D Nq.k/. This is the third item ofAX 1 Nq.

(c) The fourth item ofAX 1 Nq is only slightly more difficult, but the basic idea isthe same. By the computations (4-2),H i.DP�

Np Œ�n�x/ comes down to com-

puting I NpH ci .cLIF /, which we know is isomorphic toI NpH c

i .LIF / wheni < k � 1� Np.k/, i.e., i � Nq.k/. It is then an easy argument to show that infact the attaching map condition ofAX 1 Nq holds in this range.

(d) The first axiom also follows from these computations; onechecks that thevanishing ofH i. NP �

Np;x/ for i < 0 and fori > Np.k/ for x 2Xn�k is sufficient

to imply thatH i.DP�

Np Œ�n�x/ also vanishes fori < 0 or i sufficiently large.

We see quite clearly from these arguments precisely why the dual perversitycondition Np.k/C Nq.k/D k � 2 is necessary in order for duality to hold.


A more general duality statement, valid over principal ideal domains, wasprovided by Goresky and Siegel in [34]. However, there is an added requirementthat the spaceX be locally . Np;R/-torsion free. This means that for eachx 2Xn�k , I NpH c

k�2� Np.k/.Lx/ is R-torsion free, whereLx is the link ofx in X . The

necessity of this condition is that when working over a principal ideal domainR,the Ext terms of the universal coefficient theorem for Verdier duals must be takeninto account. If these link intersection homology groups had torsion, there wouldbe a possibly nonzero Ext term in the computation (4-2) wheni D Nq.k/C 1,due to the degree shift in the Ext term of the universal coefficient theorem. Thiswould prevent the proof thatDP�

Np Œ�n� satisfiesAX 1 Nq, so this possibility iseliminated by hypothesis. With these assumption, there result duality pairingsanalogous to those that occur for manifolds using ordinary homology withZ

coefficients. In particular, one obtains a nondegenerate intersection pairing onhomology mod torsion and a nondegenerate torsion linking pairing on torsionsubgroups. See [34] and [22] for more details.

This circle of ideas is critical in leading to the need for superperversities inthe Cappell–Shaneson superduality theorem, which we shallnow discuss.

4.2. Cappell–Shaneson superduality.The first serious application (of whichthe author is aware) of a non-GM perversity in sheaf theoretic intersection ho-mology occurs in Cappell and Shaneson’s [12], where they develop a gener-alization of the Blanchfield duality pairing of knot theory to studyL-classesof certain codimension 2 subpseudomanifolds of manifolds.Their pairing isa perfect Hermitian pairing between the perversityNp intersection homologyH�.X IP�

Np;G/ (with Np a GM perversity) andHn�1��.X IP�

Nq;G�/, whereG� isa Hermitian dual system toG and Nq satisfiesNp.k/C Nq.k/D k�1. This assuresthat Nq satisfies the GM perversity conditionNq.k/� Nq.kC1/� Nq.k/C1, but it alsoforces Nq.2/ D 1. In [26], we referred to such perversities assuperperversities,though this term was later expanded by the author to include larger classes ofperversitiesNq for which Nq.k/ may be greater thanNt.k/D k � 2 for somek.

Cappell and Shaneson worked with the sheaf version of intersection homol-ogy throughout. Notice that the Deligne sheaf remains perfectly well-defineddespiteNq being a non-GM perversity; the truncation process just starts at a higherdegree. Let us sketch how these more general perversities come into play in theCappell–Shaneson theory.

The Cappell–Shaneson superduality theorem holds in topological settings thatgeneralize those in which one studies the Blanchfield pairing of Alexander mod-ules in knot theory; see [12] for more details. The Alexandermodules are thehomology groups of infinite cyclic covers of knot complements, and one of thekey features of these modules is that they are torsion modules over the principalideal domainQŒt; t�1�. In fact, the Alexander polynomials are just the products

196 GREG FRIEDMAN

of the torsion coefficients of these modules. Similarly, theCappell–Shaneson in-tersection homology groupsH�.X IP�

Np;G/ are torsion modules overQŒt; t�1� (in

fact,G is a coefficient system with stalks equal toQŒt; t�1� and with monodromyaction determined by the linking number of a closed path withthe singular locusin X ). Now, what happens if we try to recreate the Poincare duality argumentfrom Section 4.1 in this context? For one thing, the dual of the coefficient systemoverX �X n�2 becomes the dual systemG�. More importantly, all of the Homterms in the universal coefficient theorem for Verdier duality vanish, because allmodules are torsion, but the Ext terms remain. From here, it is possible to finishthe argument, replacing all Homs with Exts, but there is one critical difference.Thanks to the degree shift in Ext terms in the universal coefficient theorem, ata pointx 2 Xn�k , H i.DP�

Np;G Œ�n�x/ vanishes not fori > k � 2� Np.k/ but fori > k�1� Np.k/, while the attaching isomorphism holds fori � k�1� Np.k/. Itfollows thatDP�

Np;G Œ�n�x is quasi-isomorphic toP�

Nq;G� , but now Nq must satisfyNp.k/C Nq.k/D k � 1.

The final duality statement that arises has the form

I NpHi.X IG/� Š Ext.I NqHn�i�1.X IG/;QŒt; t

�1�/

Š Hom.I NqHn�i�1.X IG/IQ.t; t�1/=QŒt; t�1�/;

where Np.k/C Nq.k/D k � 1, X is compact and orientable, and the last isomor-phism is from routine homological algebra. We refer the reader to [12] for theremaining technical details.

Note that this is somewhat related to our brief discussion ofthe Goresky–Siegel duality theorem. In that theorem, a special condition was added to ensurethe vanishing of the extra Ext term. In the Cappell–Shanesonduality theorem,the extra Ext term is accounted for by the change in perversity requirements, butit is important that all Hom terms vanish, otherwise there would still be a mis-match between the degrees in which the Hom terms survive truncation and thedegrees in which the Ext terms survive truncation. It might be an enlighteningexercise for the reader to work through the details.

While the Cappell–Shaneson superduality theorem generalizes the Blanch-field pairing in knot theory, the author has identified an intersection homologygeneralization of the Farber–LevineZ-torsion pairing in knot theory [21]. Inthis case, the duality statement involves Ext2 terms and requires perversitiessatisfying the duality conditionNp.k/C Nq.k/D k.

5. Subperversities and superperversities

We have already noted that King considered singular chain intersection ho-mology for perversities satisfyingNp.2/ > 0, and, more generally, he defined in


[41] a looseperversity to be an arbitrary function fromf2; 3; : : :g to Z. It is nothard to see that the PL and singular chain definitions of intersection homology(with constant coefficients) go through perfectly well withloose perversities,though we have seen that we would expect to forfeit topological invariance (andperhaps Poincare duality) with such choices. On the sheaf side, Cappell andShaneson [12] used a perversity withNp.2/ > 0 in their superduality theorem.Somewhat surprisingly, however, once we have broken into the realm of non-GM perversities, the sheaf and chain theoretic versions of intersection homologyno longer necessarily agree.

A very basic example comes by takingNp.k/<0 for somek; we will call sucha perversity asubperversity. In the Deligne sheaf construction, a subperversitywill truncate everything away and wind up with the trivial sheaf complex, whosehypercohomology groups are all0. In the chain construction, however, we haveonly made it more difficult for a chain to be allowable with respect to thekthstratum. In fact, it is shown in [22, Corollary 2.5] that the condition Np.k/ < 0

is homologically equivalent to declaring that allowable chains cannot intersectthekth stratum at all. So, for example, ifNp.k/ < 0 for all k, thenI NpH c

�.X /Š

H c�.X �X n�2/.

The discrepancy between sheaf theoretic and chain theoretic intersection ho-mology also occurs when perversities exceed the top perversity Nt.k/ D k � 2

for somek; we call such perversitiessuperperversities. To see what the issueis, let us return once again to the cone formula, which we haveseen plays thedefining local (and hence global) role in intersection homology. So long asNpis non-decreasing (and non-negative), the arguments of thepreceding sectionagain yield the sheaf-theoretic cone formula (4-1) from theDeligne construc-tion. However, the cone formula can fail in the chain versionof superperverseintersection homology.

To understand why, supposeL is a compactk � 1 pseudomanifold, so that.cL/k D v, the cone point. Recall from Section 2.4 that the cone formula comesby considering cones on allowable cycles and checking whether or not they areallowable with respect tov. In the dimensions where such cones are allowable,this kills the homology. In the dimensions where the cones are not allowable,we also cannot have any cycles intersecting the cone vertex,and the intersectionhomology reduces toI NpH c

i .cL � v/ Š I NpH ci .L � R/ Š I NpH c

i .L/, the firstisomorphism becausecL� v is homeomorphic toL�R and the second usingthe Kunneth theorem with the unstratifiedR (see [41]) or stratum-preservinghomotopy equivalence (see [23]). These arguments hold in both the PL and sin-gular chain settings. However, there is a subtle point thesearguments overlookwhen perversities exceedNt .

198 GREG FRIEDMAN

If Np is a GM perversity, thenNp.k/� k � 2 and sok � 1� Np.k/ > 0 and thecone formula guarantees thatI NpH c

0.cL/ is always isomorphic toI NpH c

0.L/. In

fact, we have already observed, in Section 3.1, that0- and1-simplices cannotintersect the singular strata. Now suppose thatNp.k/ D k � 1. Then extendingthe cone formula should predict thatI NpH c

0.cL/D 0. But, in these dimensions,

the argument breaks down. For ifx is a point incL� .cL/k�2 representing acycle inI NpSc

0.cL/, then Ncx is a1-simplex, and a quick perversity computation

shows that it is now an allowable1-simplex. However, it is not allowable as achain since@. Ncx/ has two0-simplices, one supported at the cone vertex. Thiscone vertex is not allowable. The difference between this case and the priorones is that wheni > 0 the boundary of a cone on ani-cycle is (up to sign) thati-cycle. But wheni D 0, there is a new boundary component. In the previouscomputations, this was not an issue because the1-simplex would not have beenallowable either. But now this ruins the cone formula.

In general, a careful computation shows that ifL is a compactk � 1 filteredspace andNp is any loose perversity, then the singular intersection homologycone formula becomes [41]

I NpH ci .cL/Š

8

<

:

0 if i � k � 1� Np.k/ andi ¤ 0,Z if i D 0 and Np.k/� k � 1,I NpHi.L/ if i < k � 1� Np.k/.

(5-1)

Which is the right cone formula? So when we allow superperversities withNp.k/ > Nt.k/D k �2, the cone formula (2-1) no longer holds for singular inter-section homology, and there is a disagreement with the sheaftheory, for whichthe sheaf version (4-1) of (2-1) always holds by the construction of the Delignesheaf (at least so long asNp is non-decreasing). What, then, is the “correct”version of intersection homology for superperversities (and even more generalperversities)? Sheaf theoretic intersection homology allows the use of tools suchas Verdier duality, and the superperverse sheaf intersection homology plays a keyrole in the Cappell–Shaneson superduality theorem. On the other hand, singularintersection homology is well-defined on more general spaces and allows muchmore easily for homotopy arguments, such as those used in [41; 23; 25; 27].

In [35], Habegger and Saper created a sheaf theoretic generalization of King’ssingular chain intersection homology providedNp.k/� Np.kC1/� Np.k/C1 andNp.2/ � 0. This theory satisfies a version of Poincare duality but is somewhatcomplicated. We will return to this below in Section 11.

Alternatively, a modification of the chain theory whose homology agrees withthe hypercohomology of the Deligne sheaf even for superperversities (up to theappropriate reindexing) was introduced independently by the author in [26] andby Saralegi in [54]. This chain theory has the satisfying property of maintaining


the cone formula (2-1) for completely general perversities, even those that arenot necessarily non-decreasing, while yielding the usual intersection homologygroups for GM perversities. Recently, the author has found also a generalizationof the Deligne sheaf construction that yields sheaf complexes whose hyperco-homology groups agree with the homology groups of this chaintheory and arethe usual ones for GM perversities. Of course these groups generally will notbe independent of the stratification, but they do possess Poincare duality forpseudomanifolds. Thus this theory seems to be a reasonable candidate for themost general possible intersection homology theory. We will describe this theoryand its characteristics in the following sections.

Superperversities and codimension one strata.It is a remarkable point of in-terest that the perversity issues we have been discussing provide some additionalinsight into why codimension one strata needed to be left outof the definition ofstratified pseudomanifolds used by Goresky and MacPherson (though I do notknow if it was clear that this was the issue at the time). On theone hand, if weassume thatX has a codimension one stratum and letNp.1/ D 0, then Np.1/ isgreater thanNt.1/, which we would expect to be1� 2D�1, and so we run intothe trouble with the cone formula described earlier in this section. On the otherhand, if we let Nq.1/ D Nt.1/ D �1, then we run into the trouble with negativeperversities described prior to that. In this latter case, the Deligne sheaf is alwaystrivial, yielding only trivial sheaf intersection homology, so there can be no non-trivial Poincare duality via the sheaf route (note thatNp.1/ D 0 and Nq.1/ D �1

are dual perversities atk D 1, so any consideration of duality involving the oneperversity would necessarily involve the other). Similarly, there is no dualityin the chain version since, for example, ifX n � X n�1 is S1 � pt then easycomputations shows thatI NqH1.X /ŠH c

1.S1 � pt/D 0, while I NpH0.X /Š Z.

Note that the first computation shows that we have also voidedthe stratificationindependence of intersection homology.

One of the nice benefits of our (and Saralegi’s) “correction”to chain-theoreticintersection homology is that it allows one to include codimension one strata andstill obtain Poincare duality results. In general, though, the stratification inde-pendence does need to be sacrificed. One might argue that thisis the preferredtrade-off, since one might wish to use duality as a tool to study spacestogetherwith their stratifications.

6. “Correcting” the definition of intersection chains

As we observed in the previous section, ifNp is a superperversity (i.e.,Np.k/>k�2 for somek), then the Deligne sheaf version of intersection homology andthe chain version of intersection homology need no longer agree. Modifications

200 GREG FRIEDMAN

of the chain theory to correct this anomaly were introduced by the author in[26] and by Saralegi in [54], and these have turned out to provide a platformfor the extension of other useful properties of intersection homology, includingPoincare duality. These modifications turn out to be equivalent, as proven in[28]. We first present the author’s version, which is slightly more general in thatit allows for the use of local coefficient systems onX �X n�1.

As we saw in Section 5, the discrepancy between the sheaf coneformula andthe chain cone formulas arises because the boundary of a1-chain that is the coneon a0-chain has a0-simplex at the cone point. So to fix the cone formula, it isnecessary to find a way to make the extra0 simplex go away. This is preciselywhat both the author’s and Saralegi’s corrections do, though how they do it isdescribed in different ways.

The author’s idea, motivated by the fact that Goresky–MacPherson perversityintersection chains need only have their coefficients well-defined onX �X n�2,was to extend the coefficientsG on X �X n�1 (now allowing codimension onestrata) to astratified coefficient systemby including a “zero coefficient system”onX n�1. Together these are denoteG0. Then a coefficient on a singular simplex� W�i!X is defined by a lift of� j��1.X �X n�1/ to the bundleG onX �X n�1

and by a “lift” of � j��1.X n�1/ to the0 coefficient system overX n�1. Boundaryfaces then inherit their coefficients from the simplices they are boundaries ofby restriction. A simplex has coefficient0 if its coefficient lift is to the zerosection over all of�i . In the PL setting, coefficients of PL simplices are definedsimilarly. In principle, there is no reason the coefficient system overX n�1 mustbe trivial, and one could extend this definition by allowing different coefficientsystems on all the strata ofX ; however, this idea has yet to be investigated.

With this coefficient systemG0, the intersection chain complexI NpS�.X IG0/

is defined exactly as it is with ordinary coefficients — allowability of simplices isdetermined by the same formula, and chains are allowable if each simplex witha nonzero coefficient in the chain is allowable. So what has changed? The subtledifference is that if a simplex that is in the boundary of a chain has support inX n�1, then that boundary simplex must now have coefficient0, since that is theonly possible coefficient for simplices inX n�1; thus such boundary simplicesvanish and need not be tested for allowability. This simple idea turns out to beenough to fix the cone formula.

Indeed, let us reconsider the example of a pointx in cL � .cL/k�2, to-gether with a coefficient lift toG, representing a cycle inI NpSc

0.cLIG0/, where

Np.k/ D k � 1. As before, Ncx is a 1-simplex, and it is allowable. Previously,Ncx was not, however, allowableas a chainsince the component of@. Ncx/ in thecone vertex was not allowable. However, if we consider the boundary of Ncx inI NpSc

0.cLIG0/, then the simplex at the cone point vanishes because it must have


a zero coefficient there. Thus allowability is not violated by Ncx; it is now anallowablechain.

A slightly more detailed computation (see [26]) shows that,in fact,

I NpH ci .cLk�1IG0/Š

�

0 if i � k � 1� Np.k/,I NpH c

i .LIG0/ if i < k � 1� Np.k/,(6-1)

i.e., we recover the cone formula, even ifNp.k/ > k � 2.Another pleasant feature ofI NpH�.X IG0/ is that if Np does happen to be a

GM perversity andX has no codimension one strata, thenI NpH�.X IG0/ Š

I NpH�.X IG/, the usual intersection homology. In fact, this follows from ourdiscussion in Section 3.1, where we noted that ifNp is a GM perversity thenno allowablei-simplices intersectX n�2 in either the interiors of theiri facesor the interiors of theiri � 1 faces. Thus no boundary simplices can lie en-tirely in X n�2 and canceling of boundary simplices due to the stratified co-efficient system does not occur. ThusI NpH�.X IG0/ legitimately extends theoriginal Goresky–MacPherson theory. Furthermore, working with this “cor-rected” cone formula, one can show that the resulting intersection homologygroupsI NpH 1

� .X IG0/ agree on topological stratified pseudomanifolds (modulothe usual reindexing issues) with the Deligne sheaf hypercohomology groups(and similarly with compact supports), assuming thatNp.2/ � 0 and that Np isnon-decreasing. This was proven in [26] under the assumption that Np.2/D 0 or1 and that Np.k/� Np.kC1/� Np.k/C1, but the more general case follows from[22].

Thus, in summary,I NpH�.X IG0/ satisfies the cone formula, generalizes in-tersection homology with GM perversities, admits codimension one strata, andagrees with the Deligne sheaf for the superperversities we have considered upto this point. It turns out that stratified coefficients also permit useful results foreven more general contexts.

REMARK 6.1. A similar idea for modifying the definition of intersection ho-mology for non-GM perversities occurs in the unpublished notes of MacPherson[44]. There, only locally-finite chains inX � X n�1 are considered, but theirclosures inX are used to determine allowability.

7. General perversities

We have now seen that stratified coefficientsG0 allow us to recover the coneformula (6-1) both whenNp is a GM perversity and when it is a non-decreasingsuperperversity. How far can we push this? The answer turns out to be “quitefar!” In fact, the cone formula will hold if Np is completely arbitrary. Recallthat we have defined a stratum ofX to be a connected component of anyXk D

202 GREG FRIEDMAN

X k �X k�1. For a stratified pseudomanifold, possibly with codimension onestrata, we define ageneral perversityNp on X to be a function

Np W fsingular strata of Xg ! Z:

Then a singular simplex� W�i! X is Np-allowable if

��1.Z/� fi � codim.Z/C Np.Z/ skeleton of�ig

for each singular stratumZ of X . Even in this generality, the cone formula(6-1) holds forI NpH c

� .cLk�1IG0/, replacing Np.k/ with Np.v/, wherev is thecone vertex.

Such general perversities were considered in [44], following their appearancein the realm of perverse sheaves (see [4] and Section 8.2, below), and they appearin the work of Saralegi on intersection differential forms [53; 54]. They alsoplay an important role in the intersection homology Kunneth theorem of [28],which utilizes “biperversities” in which the setXk � Yl � X � Y is given aperversity value depending onNp.k/ and Nq.l/ for two perversitiesNp; Nq onX andY , respectively; see Section 9.

In this section, we discuss some of the basic results on intersection homologywith general perversities, most of which generalize the known theorems for GMperversities. We continue, for the most part, with the chaintheory point ofview. In Section 8, we will return to sheaf theory and discusssheaf-theoretictechniques for handling general perversities.

REMARK 7.1. One thing that we can continue to avoid in defining generalperversities is assigning perversity values to regular strata (those inX �X n�1)and including this as part of the data to check for allowability. The reason is asfollows: If Z is a regular stratum, the allowability conditions for a singular i-simplex� would include the condition that��1.Z/ lie in theiC Np.Z/ skeletonof �i . If Np.Z/� 0, then this is true of any singulari-simplex, and if Np.Z/ < 0,then this would imply that the singular simplex must not intersectZ at all, sinceX n�1 is a closed subset ofX . Thus there are essentially only two possibilities.The caseNp.Z/� 0 is the default that we work with already (without explicitlychecking the condition that would always be satisfied on regular strata). On theother hand, the caseNp.Z/ < �1 is something of a degeneration. IfNp.Z/ < 0

for all regular strata, then all singular chains must be supported inX n�1 andsoI NpS�.X IG0/D 0. If there are only some regular strata such thatNp.Z/ < 0,then, lettingX C denote the pseudomanifold that is the closure of the unionof the regular strataZ of X such that Np.Z/ � 0, we haveI NpH�.X IG0/ Š

I NpH�.XCIG0jX C/. We could have simply studied intersection homology on

X C in the first place, so we get nothing new. Thus it is reasonableto concernourselves only with singular strata in defining allowability of simplices.


This being said, there are occasional situations where it isuseful in technicalformulae to assume thatNp.Z/ is defined for all strata. This comes up, for exam-ple, in [28], where we define perversities on product strataZ1�Z2 �X1�X2

using formulas such asQf Np; Nqg.Z1 � Z2/ D Np.Z1/C Nq.Z2/ for perversitiesNp; Nq. HereZ1 � Z2 may be a singular stratum, for example, even ifZ1 isregular butZ2 is singular. The formula has the desired consequence in [28]bysetting Np.Z1/ D 0 for Z1 regular, and this avoids having to write out severalcases.

Efficient perversities. It turns out that such generality contains a bit of overkill.In [22], we define a general perversityNp to beefficientif

�1� Np.Z/� codim.Z/� 1

for each singular stratumZ�X . Given a generalNp, we define itsefficientizationLp as

Lp.Z/D

8

<

:

codim.Z/� 1 if Np.Z/� codim.Z/� 1,Np.Z/ if 0� Np.Z/� codim.Z/� 2,�1 if Np.Z/� �1.

It is shown in [22, Section 2] thatI NpH�.X IG0/ Š I LpH�.X IG0/. Thus it isalways sufficient to restrict attention to the efficient perversities.

Efficient perversities and interiors of simplices. Efficient perversities have anice feature that makes them technically better behaved than the more generalperversities. If Np is a perversity for whichNp.Z/� codim.Z/ for some singularstratumZ, then anyi-simplex � will be Np-allowable with respect toZ. Inparticular,Z will be allowed to intersect the image under� of the interior of�i . As such,��1.X � X n�1/ could potentially have an infinite number ofconnected components, and a coefficient of� might lift each component to adifferent branch ofG, even ifG is a constant system. This could potentiallylead to some pathologies, especially when considering intersection chains fromthe sheaf point of view. However, ifNp is efficient, then for aNp-allowable� wemust have��1.X �X n�1/ within the i � 1 skeleton of�i . Hence assigning acoefficient lift value to one point of the interior of�i determines the coefficientvalue at all points (on��1.X �X n�1/ by the unique extension of the lift andon ��1.X n�1/, where it is0). This is technically much simpler and makes thecomplex of chains in some sense smaller.

In [28], the complexI NpS�.X IG0/ was defined with the assumption that this“unique coefficient” property holds, meaning that a coefficient should be deter-mined by its lift at a single point. However, as noted in [28, Appendix], even forinefficient perversities, this does not change the intersection homology. So weare free to assume all perversities are efficient, without loss of any information

204 GREG FRIEDMAN

(at least at the level of quasi-isomorphism), and this provides a reasonable wayto avoid the issue entirely.

7.1. Properties of intersection homology with general perversities and strat-ified coefficients. One major property that we lose in working with generalperversities and stratified coefficients is independence ofstratification. However,most of the other basic properties of intersection homologysurvive, includingPoincare duality, some of them in even a stronger form than GM perversitiesallow.

Basic properties. SupposeX n is a topological stratified pseudomanifold, pos-sibly with codimension one strata, letG be a coefficient system onX �X n�1,and let Np be a general perversity. What properties doesI NpH�.X IG0/ possess?

For one thing, the most basic properties of intersection homology remainintact. It is invariant under stratum-preserving homotopyequivalences, and itpossesses an excision property, long exact sequences of thepair, and Mayer–Vietoris sequences. The Kunneth theorem when one term is an unstratifiedmanifold M holds true (i.e.,I NpSc

�.X �M I .G � G0/0/ is quasi-isomorphic to

I NpSc�.X IG0/˝Sc

�.M IG0

0/). There are versions of this intersection homology

with compact supports and with closed supports. AndU!I NpS�.X;X� NU IG0/

sheafifies to a homotopically fine sheaf whose hypercohomology groups recoverthe intersection homology groups, up to reindexing. It is also possible to workwith PL chains on PL pseudomanifolds. For more details, see [26; 22].

Duality. Let us now discuss Poincare duality in our present context.

THEOREM 7.2 (POINCARE DUALITY ). If F is a field8, X is an F -orientedn-dimensional stratified pseudomanifold, and NpC Nq D Nt (meaning thatNp.Z/CNq.Z/D codim.Z/� 2 for all singular strataZ), then

I NpH 1

i .X IF0/Š Hom.I NqH cn�i.X IF0/;F /:

For compact orientable PL pseudomanifolds without codimension one strataand with GM perversities, this was initially proven in [32] via a combinatorialargument; a proof extending to the topological setting using the axiomatics ofthe Deligne sheaf and Verdier duality was obtained in [33]. This Verdier dualityproof was extended to the current setting in [22] using a generalization of theDeligne sheaf that we will discuss in the following section.It also follows fromthe theory of perverse sheaves [4]. Recent work of the authorand Jim McClurein [31] shows that intersection homology Poincare duality can be proven usinga cap product with an intersection homology orientation class by analogy to

8Recall that even in the Goresky–MacPherson setting, duality only holds, in general,with field coefficients.


the usual proof of Poincare duality on manifolds (see, [36], for example). Aslightly more restrictive statement (without proof) of duality for general perver-sities appears in the unpublished lecture notes of MacPherson [44] as far backas 1990.

As is the case for classical intersection homology, more general duality state-ments hold. These can involve local-coefficient systems, non-orientable pseudo-manifolds, and, ifX is locally . Np;R/-torsion freefor the principal ideal domainR, then there are torsion linking and mod torsion intersection dualities overR.For complete details, see [22].

Pseudomanifolds with boundary and Lefschetz duality.General perversitiesand stratified coefficients can also be used to give an easy proof of a Lefschetzversion of the duality pairing, one for whichX is a pseudomanifold with bound-ary:

DEFINITION 7.3. An n-dimensionalstratified pseudomanifold with boundaryis a pair.X; @X / such thatX � @X is ann-dimensional stratified pseudoman-ifold and theboundary@X is ann � 1 dimensional stratified pseudomanifoldpossessing a neighborhood inX that is stratified homeomorphic to@X � Œ0; 1/,whereŒ0; 1/ is unstratified and@X � Œ0; 1/ is given the product stratification.

REMARK 7.4. A pseudomanifold may have codimension one strata that arenot part of a boundary, even if they would be considered part of a boundaryotherwise. For example, letM be a manifold with boundary@M (in the usualsense). If we considerM to be unstratified, then@M is the boundary ofM .However, if we stratifyM by the stratificationM � @M , then@M is not aboundary ofM as a stratified pseudomanifold, and in this caseM is a stratifiedpseudomanifold without boundary.

We can now state a Lefschetz duality theorem for intersection homology ofpseudomanifolds with boundary.

THEOREM 7.5 (LEFSCHETZ DUALITY). If F is a field, X is a compactF -orientedn-dimensional stratified pseudomanifold, and NpC Nq D Nt (meaning thatNp.Z/C Nq.Z/D codim.Z/� 2 for all singular strataZ), then

I NpHi.X IF0/Š Hom.I NqHn�i.X; @X IF0/;F /:

This duality also can be extended to include local-coefficient systems, non-compact or non-orientable pseudomanifolds, and, ifX is locally . Np;R/-torsionfree for the principal ideal domainR, then there are torsion linking and modtorsion intersection dualities overR.

In fact, in the setting of intersection homology with general perversities,this Lefschetz duality follows easily from Poincare duality. To see this, let

206 GREG FRIEDMAN

OX DX [@X Nc@X , the space obtained by adjoining toX a cone on the boundary(or, equivalently, pinching the boundary to a point). Letv denote the ver-tex of the cone point. LetNp�, NqC be the dual perversities onOX such thatNp�.Z/D Np.Z/ and NqC.Z/D Nq.Z/ for each stratumZ of X , Np�.v/D�2, andNqC.v/D n. Poincare duality gives a duality isomorphism betweenI Np�H�. OX /

andI NqCH�. OX /. But now we simply observe thatI Np�H�. OX / Š I Np�H�. OX �

v/ Š I NpH�.X /, because the perversity condition atv ensures that no singularsimplex may intersectv. On the other hand, sinceI NqCH�.c@X /D 0 by the coneformula,I NqCH�. OX /Š I NqCH�. OX ; Nc@X / by the long exact sequence of the pair,but I NqCH�. OX ; Nc@X /Š I NqCH�.X; @X /Š I NqH�.X; @X / by excision.

Notice that general perversities are used in this argument even if Np and Nq areGM perversities.

PL intersection pairings. As in the classical PL manifold situation, the dualityisomorphism of intersection homology arises out of a more general pairing ofchains. In [32], Goresky and MacPherson defined the intersection pairing of PLintersection chains in a PL pseudomanifold as a generalization of the classicalmanifold intersection pairing. For manifolds, the intersection pairing is dualto the cup product pairing in cohomology. Given a ringR and GM perversitiesNp; Nq; Nr such thatNpC Nq� Nr , Goresky and MacPherson constructed an intersectionpairing

I NpH ci .X IR/˝ I NqH c

j .X IR/! INr H ciCj�n.X IR/:

This pairing arises by pushing cycles into a stratified version of general positiondue to McCrory [47] and then taking chain-theoretic intersections.

The Goresky–MacPherson pairing is limited in that aNp-allowable chain anda Nq-allowable chain can be intersected only if there is a GM perversity Nr suchthat Np C Nq � Nr . In particular, we must haveNp C Nq � t . This is more thansimply a failure of the intersection of the chains to be allowable with respectto a GM perversity — if Np C Nq 6� Nt , there are even technical difficulties withdefining the intersection product in the first place. See [22,Section 5] for an indepth discussion of the details.

If we work with stratified coefficients, however, the problems mentioned inthe preceding paragraphs can be circumvented, and we obtainpairings

I NpHi.X IR0/˝ I NqHj .X IR0/! INr HiCj�n.X IR0/

for anygeneral perversities such thatNpC Nq � Nr .Goresky and MacPherson extended their intersection pairing to topological

pseudomanifolds using sheaf theory [33]. This can also be done for generalperversities and stratified coefficients, but first we must revisit the Deligne sheafconstruction. We do so in the next section.


A new approach to the intersection pairing via intersectioncohomology cupproducts is presently being pursued by the author and McClure in [31].

Further applications. Some further applications of general perversity intersec-tion homology will be discussed below in Section 9.

8. Back to sheaf theory

8.1. A generalization of the Deligne construction. Intersection chains withstratified coefficients were introduced to provide a chain theory whose homologyagrees with the hypercohomology of the Deligne sheaf whenNp is a superper-versity, in particular whenNp.2/ > 0 or whenX has codimension one strata.However, whenNp is a general perversity, our new chain formulation no longagrees with the Deligne construction. For one thing, we knowthat if Np is evernegative, the Deligne sheaf is trivial. The classical Deligne construction alsohas no mechanism for handling perversities that assign different values to strataof the same codimension, and, even if we restrict to less general perversities,any decrease in perversity value at a later stage of the Deligne process willtruncate away what might have been vital information comingfrom an earlierstage. Thus, we need a generalization of the Deligne processthat incorporatesgeneral perversities and stratified coefficients. One method was provided by theauthor in [22], and we describe this now.

The first step is to modify the truncation functor to be a bit more picky. Ratherthan truncating a sheaf complex in the same degree at all stalks, we truncate morelocally. This new truncation functor is a further generalization of the “truncationover a closed subset” functor presented in [33, Section 1.14] and attributed toDeligne; that functor is used in [33, Section 9] to study extensions of Verdierduality pairings in the context of intersection homology with GM perversities.Our construction is also related to the “intermediate extension” functor in thetheory of perverse sheaves; we will discuss this in the next subsection.

DEFINITION 8.1. LetA� be a sheaf complex onX , and letF be a locally-finitecollection of subsets ofX . Let jFj D [V 2FV . Let P be a functionF ! Z.Define the presheafT F

�PA� as follows. IfU is an open set ofX , let

T F

�PA

�.U /D

�

� .U IA�/ if U \ jFj D?,� .U I ��inffP.V /jV 2F;U \V ¤?gA

�/ if U \ jFj ¤?.

Restriction is well-defined because ifm < n then there is a natural inclusion��mA�Œ ��nA

�.Let thegeneralized truncation sheaf�F

�PA� be the sheafification ofT F

�PA�.

For mapsf W A� ! B� of sheaf complexes overX , we can define�F

�Pf

in the obvious way. In fact,T F

�Pf is well-defined by applying the ordinary

208 GREG FRIEDMAN

truncation functors on the appropriate subsets, and we obtain �F

�Pf again by

passing to limits in the sheafification process.

Using this truncation, we can modify the Deligne sheaf.

DEFINITION 8.2. LetX be ann-dimensional stratified pseudomanifold, possi-bly with codimension one strata, letNp be a general perversity, letG be a coef-ficient system onX �X n�1, and letO be the orientation sheaf onX �X n�1.Let Xk stand also for the set of strata of dimensionk. Then we define thegeneralized Deligne sheafas9

Q�

Np;G D �X0

� NpRin� : : : �Xn�1

� Np Ri1�.G˝O/:

If Np is a GM perversity, then it is not hard to show directly thatQ�

Np;G is quasi-isomorphic to the usual Deligne sheafP�

Np;G . Furthermore, it is shown in [22]thatQ�

Np;G is quasi-isomorphic to the sheaf generated by the presheaf

U ! I NpSn��.X;X � NU IG0/;

and soH�.Q�

Np;G/ Š I NpH 1n��.X IG0/ and similarly for compact supports. It is

also true, generalizing the Goresky–MacPherson case, thatif Np C Nq D Nt , thenQ�

Np andQ�

Nq are appropriately Verdier dual, leading to the expected Poincareand Lefschetz duality theorems. Furthermore, for any general perversities suchthat NpC Nq � Nr , there are sheaf pairingsQ�

Np˝Q�

Nq !Q�

Nr that generalize the PLintersection pairing. IfNpC Nq � Nt , there is also a pairingQ�

Np ˝Q�

Nq ! D�

XŒ�n�,

whereD�

XŒ�n� is the shifted Verdier dualizing complex onX . See [22] for the

precise statements of these results.

8.2. Perverse sheaves.The theory of perverse sheaves provided, as far backas the early 1980s, a context for the treatment of general perversities. To quoteBanagl’s introduction to [2, Chapter 7]:

In discussing the proof of the Kazhdan–Lusztig conjecture,Beilinson,Bernstein and Deligne discovered that the essential image of the categoryof regular holonomicD-modules under the Riemann–Hilbert correspon-dence gives a natural abelian subcategory of the nonabelianbounded con-structible derived category [of sheaves] on a smooth complex algebraicvariety. An intrinsic characterization of this abelian subcategory was ob-tained by Deligne (based on discussions with Beilinson, Bernstein, andMacPherson), and independently by Kashiwara. It was then realized thatone still gets an abelian subcategory if the axioms of the characterization

9This definition differs from that in [22] by the orientation sheafO— see Remark4.1 on page 190. For consistency, we also change notation slightly to includeG as asubscript rather than as an argument.


are modified to accommodate an arbitrary perversity function, with theoriginal axioms corresponding to the middle perversity. The objects ofthese abelian categories were termedperverse sheaves. . .

Thus, the phrase “perverse sheaves” refers to certain subcategories, indexed byvarious kinds of perversity functions, of the derived category of bounded con-structible sheaf complexes on a spaceX . The general theory of perverse sheavescan handle general perversities, though the middle perversities are far-and-awaythose most commonly encountered in the literature (and, unfortunately, manyexpositions restrict themselves solely to this case). The remarkable thing aboutthese categories of perverse sheaves is that they are abelian, which the derivedcategory is not (it is only “triangulated”).10 The Deligne sheaf complexes onthe various strata ofX (and with appropriate coefficients systems) turn out tobe the simple objects of these subcategories.

The construction of perverse sheaves is largely axiomatic,grounded in a num-ber of quite general categorical structures. It would take us well too far afieldto provide all the details. Rather, we provide an extremely rough sketch of theideas and refer the reader to the following excellent sources: [4], [40, ChapterX], [2, Chapter 7], [5], and [20, Chapter 5]. For a more historical account, thereader should see [43].

The starting point for any discussion of perverse sheaves isthe notion ofT -structures. Very roughly, aT -structure on a triangulated categoryD is a pair ofsubcategories.D�0;D�0/ that are complementary, in the sense that for anyS

in D, there is a distinguished triangle

S1! S ! S2;

with S1 2 D�0 andS2 in D�0. Of course there are a number of axioms thatmust be satisfied and that we will not discuss here. The notation reflects thecanonicalT -structure that occurs on the derived category of sheaves ona spaceX : D�0.X / is defined to be those sheaf complexesS� such thatHj .S�/D 0

for j > 0, andD�0.X / is defined to be those sheaf complexesS� such thatHj .S�/D 0 for j < 0. HereH�.S�/ denotes the derived cohomology sheaf ofthe sheaf complexS�, such thatH�.S�/x DH �.S�

x/.The heart (or core) of a T -structure is the intersectionD�0 \D�0. It is

always an abelian category. In our canonical example, the heart consists of thesheaf complexes with nonvanishing cohomology only in degree 0. In this case,the heart is equivalent to the abelian category of sheaves onX . Already from

10There is an old joke in the literature that perverse sheaves are neither perverse norsheaves. The first claim reflects the fact that perverse sheaves form abelian categories,which are much less “perverse” than triangulated categories. The second reflects simplythe fact that perverse sheaves are actually complexes of sheaves.

210 GREG FRIEDMAN

this example, we see how truncation might play a role in providing perversesheaves — in fact, for the sheaf complexS�, the distinguished triangle in thisexample is provided by

��0S�! S

�! ��0S�:

Furthermore, this example can be modified easily by shiftingthe truncation de-gree from0 to any other integerk. ThisT -structure is denoted by

.D�k.X /;D�k.X //:

The next important fact aboutT -structures is that ifX is a space,U is anopen subspace,F DX�U , andT -structures satisfying sufficient axioms on thederived categories of sheaves onU andF are given, they can be “glued” to pro-vide aT -structure on the derived category of sheaves onX . The idea the readershould have in mind now is that of gluing together sheaves truncated at a certaindimension onU and at another dimension onF . This then starts to look a bitlike the Deligne process. In fact, letP be a perversity11 on the two stratum spaceX �F , and let.D�P.U /.U /;D�P.U /.U // and.D�P.F /.F /;D�P.F /.F // beT -structures onU andF . Then theseT -structures can be glued to form aT -structure onX , denoted by.PD�0;PD�0/.

It turns out that the subcategoriesPD�0 and PD�0 can be described quiteexplicitly. If i W U ŒX andj W F ŒX are the inclusions, then

PD�0 D

�

S� 2DC.X /

ˇ

ˇ

ˇ

ˇ

Hk.i�

S�/D 0 for k > P .U /

Hk.j�

S�/D 0 for k > P .F /

�

;

PD�0 D

�

S� 2DC.X /

ˇ

ˇ

ˇ

ˇ

Hk.i�

S�/D 0 for k < P .U /

Hk.j !

S�/D 0 for k < P .F /

�

:

If S� is in the heart of thisT -structure, we say it isP -perverse.More generally, ifX is a space with a variety of singular strataZ andP is

a perversity on the stratification ofX , then it is possible to glueT -structuresinductively to obtain the category ofP -perverse sheaves. IfjZ WZŒX are theinclusions, then theP -perverse sheaves are those which satisfyHk.j �

ZS�/D 0

for k > P .Z/ andHk.j !ZS�/D 0 for k < P .Z/.

These two conditions turn out to be remarkably close to the conditions forS�

to satisfy the Deligne sheaf axiomsAX 1. In fact, the conditionHk.j �

ZS�/D 0

for k > P .Z/ is precisely the third axiom. The conditionHk.j !ZS�/ D 0 for

11The reason we useP here for a perversity, departing from both our own notation,above, and from the notation in most sources on perverse sheaves (in particular [4]) isthat when we use perverse sheaf theory, below, to recover intersection homology, therewill be a discrepancy between the perversityP for perverse sheaves and the perversityNp for the Deligne sheaf.


k < P .Z/ implies that the local attaching map is an isomorphism up to degreeP .Z/ � 2; see [6, page 87]. Notice that this is a less strict requirement thanthat for the Deligne sheaf. Thus, Deligne sheaves are perverse sheaves, but notnecessarily vice versa.

The machinery developed in [4] also contains a method for creating sheafcomplexes that satisfy the intersection homology axiomsAX 1, though again itis more of an axiomatic construction than the concrete construction provided inSection 8.1. LetU � X be an open subset ofX that is a union of strata, leti WU ŒX be the inclusion, and letS� be aP -perverse sheaf onU . Then thereis defined in [4] the “intermediate extension functor”i!� such thati!�S� is theunique extension in the category ofP -perverse sheaves ofS� to X (meaningthat the restriction ofi!�S� to U is quasi-isomorphic toS�) such that for eachstratumZ � X �U and inclusionj W ZŒ X ,we haveHk.j �i!�S

�/D 0 fork � P .Z/ andHk.j !i!�S

�/ D 0 for k � P .Z/. We refer the reader to [4,Section 1.4] or [20, Section 5.2] for the precise definition of the functori!�.

In particular, suppose we letU DX�X n�1, thatS� is just the local systemG,and thatNp is a general perversity onX . The sheafG is certainlyP -perverse onUwith respect to the perversityP .U /D 0. Now letP .Z/D Np.Z/C1. It followsthat for each singular stratum inclusionj WZŒ X , we haveHk.j �i!�G/D 0

for k > Np.Z/ andHk.j !i!�G/ D 0 for k � Np.Z/C 1. In the presence of thefirst condition, the second condition is equivalent to the attaching map beingan isomorphism up through degreeNp.Z/; see [6, page 87]. But, accordingto the axiomsAX 1, these conditions are satisfied by the perversityNp Delignesheaf, which is also easily seen to beP -perverse. Thus, sincei!�G is the uniqueextension ofG with these properties,i!�G is none other than the Deligne sheaf(up to quasi-isomorphism)! Thus we can think of the Deligne process providedin Section 8.1 as a means to provide a concrete realization ofi!�G.

9. Recent and future applications of general perversities

Beyond extending the results of intersection homology withGM perversities,working with general perversities makes possible new results that do not existin “classical” intersection homology theory. For example,we saw in Sections7 and 8 that general perversities permit the definition of PL or sheaf-theoreticintersection pairings with no restrictions on the perversities of the intersectionhomology classes being intersected. In this section, we review some other recentand forthcoming results made possible by intersection homology with generalperversities.

212 GREG FRIEDMAN

Kunneth theorems and cup products.In [28], general perversities were usedto provide a very general Kunneth theorem for intersection homology. Somespecial cases had been known previously. King [41] showed that for any looseperversityI NpH c

�.M �X /ŠH�.C

c�.M /˝I NpC c

�.X // whenX is a pseudoman-

ifold, M is an unstratified manifold, and.M �X /i DM �X i . Special cases ofthis result were proven earlier by Cheeger [16], Goresky andMacPherson [32;33], and Siegel [55]. In [18], Cohen, Goresky, and Ji provided counterexamplesto the existence of a general Kunneth theorem for a single perversity and showedthatI NpH c

�.X �Y IR/ŠH�.I

NpC c�.X IR/˝I NpC c

�.Y IR// for pseudomanifolds

X andY and a principal ideal domainR provided either that

(a) Np.a/C Np.b/� Np.aC b/� Np.a/C Np.b/C 1 for all a andb, or that(b) Np.a/C Np.b/� Np.aCb/� Np.a/C Np.b/C2 for all a andb and eitherX or

Y is locally . Np;R/-torsion free.

The idea of [28] was to ask a broader question: for what perversities onX �Y isthe intersection chain complex quasi-isomorphic to the productI NpC c

� .X IR0/˝

I NqC c�.X IR0/? This question encompasses the Cohen–Goresky–Ji Kunneth the-

orem and the possibility of both GM and non-GM perversitiesNp; Nq. However,in order to avoid the fairly complicated conditions on a single perversity foundby Cohen, Goresky, and Ji, it is reasonable to consider general perversities onX � Y that assign to a singular stratumZ1 �Z2 a value depending onNp.Z1/

and Nq.Z2/. Somewhat surprisingly, there turn out to be many perversities onX � Y that provide the desired quasi-isomorphism. The main result of [28]is the following theorem. The statement is reworded here to account for themost general case (see [28, Theorem 3.2, Remark 3.4, Theorem5.2]), while thestatement in [28] is worded to avoid overburdening the reader too much withdetails of stratified coefficients, which play a minimal rolethat paper.

THEOREM 9.1. If R is a principal ideal domain andNp and Nq are generalperversities, thenIQH c

�.X � Y IR0/ Š H�.I

NpC c�.X IR0/˝ I NqC c

�.Y IR0// if

the following conditions hold:

(a) Q.Z1 �Z2/D Np.Z1/ if Z2 is a regular stratum ofY andQ.Z1 �Z2/D

Nq.Z2/ if Z1 is a regular stratum ofX .(b) For each pairZ1�Z2 such thatZ1 andZ2 are each singular strata, either

(i) Q.Z1 �Z2/D Np.Z1/C Nq.Z2/, or

(ii) Q.Z1�Z2/D Np.Z1/C Nq.Z2/C 1, or

(iii) Q.Z1 �Z2/D Np.Z1/C Nq.Z2/C 2 and the torsion product

I NpHcodim.Z1/�2� Np.Z1/.L1IR0/� I NqHcodim.Z2/�2� Nq.Z2/.L2IR0/

is zero, whereL1;L2 are the links ofZ1;Z2 in X;Y , respectively, andcodimrefers to codimension inX or Y , as appropriate.


Furthermore, if these conditions are not satisfied, thenIQH c�.X � Y IR0/ will

not equalH�.INpC c

� .X IR0/˝ I NqC c� .Y IR0// in general.

Of course the torsion condition in (iii) will be satisfied automatically if R isa field or if X or Y is locally ( Np;R)- or (Nq;R/-torsion free. Note also thatit is not required that a consistent choice among the above options be madeacross all products of singular strata — for each suchZ1 �Z2 one can chooseindependently which perversity to use from among options (i), (ii), or, assumingthe hypothesis, (iii). The theorem can also be generalized further to includestratified local coefficient systems onX or Y ; we leave the details to the reader.

This Kunneth theorem has opened the way toward other results in intersec-tion homology, including the formulation by the author and Jim McClure ofan intersection cohomology cup product over field coefficients that they ex-pect to be dual to the Goresky–MacPherson intersection pairing. There doesnot seem to have been much past research done on or with intersection co-homology in the sense of the homology groups of cochainsI NpC �.X IR0/ D

Hom.I NpC c�.X IR0/IR/. One important reason would seem to be the prior lack

of availability of a geometric cup product. A cup product using the Alexander–Whitney map is unavailable in intersection homology since it does not preservethe admissibility conditions for intersection chains — in other words, breakingchains into “front p-faces and back q-faces” (see [49, Section 48]) might destroyallowability of simplices. However, there is another classical approach to the cupproduct that can be adapted to intersection cohomology, provided one has anappropriate Kunneth theorem. Forordinary homology, this alternative approachis to define a diagonal map (with field coefficients) as the composite

H c�.X /!H c

�.X �X /

Š

H c�.X /˝H c

�.X /;

where the first map is induced by the geometric diagonal inclusion map andthe second is the Eilenberg–Zilber shuffle product, which isan isomorphismby the ordinary Kunneth theorem with field coefficients (note that the shuffleproduct should have better geometric properties than the Alexander–Whitneymap because it is really just Cartesian product). The appropriate Hom dual ofthis composition yields the cup product. This process suggests doing somethingsimilar in intersection homology with field coefficients, and indeed the Kunneththeorem of [28] provides the necessary right-hand quasi-isomorphism in a dia-gram of the form

INsH c�.X IF0/! IQH c

�.X �X IF0/

Š

I NpH c�.X IF0/˝ I NqH c

�.X IF0/:

When NpC Nq � Nt C Ns, there results a cup product

I NpH �.X IF0/˝ I NqH �.X IF0/! INsH �.X IF0/:

214 GREG FRIEDMAN

The intersection Kunneth theorem also allows for a cap product of the form

I NpH i.X IF0/˝ INsH cj .X IF0/! I NqH c

j�i.X IF0/

for any fieldF and any perversities satisfyingNpC Nq� NtCNs. This makes possiblea Poincare duality theorem for intersection (co)homology given by cap productswith a fundamental class inIN0Hn.X IF0/. For further details and applications,the reader is urged to consult [31].

Perverse signatures.Right from its beginnings, there has been much interestand activity in using intersection homology to define signature (index) invari-ants and bordism theories under which these signatures are preserved. Sig-natures first appeared in intersection homology in [32] associated to the sym-metric intersection pairings onI NmH2n.X

4nIQ/ for spacesX with only strataof even codimension, such as complex algebraic varieties. The condition onstrata of even codimension ensures thatI NmH2n.X

4nIQ/ Š I NnH2n.X4nIQ/

so that this group is self-dual under the intersection pairing. These ideas wereextended by Siegel [55] to the broader class of Witt spaces, which also satisfyI NmH2n.X

4nIQ/ Š I NnH2n.X4nIQ/. In addition, Siegel developed a bordism

theory of Witt spaces, which he used to construct a geometricmodel forko-homology at odd primes. Further far reaching generalizations of these signa-tures have been studied by, among others and in various combinations, Banagl,Cappell, Libgober, Maxim, Shaneson, and Weinberger [1; 3; 10; 11; 9].

Signatures on singular spaces have also been studied analytically via L2-cohomology andL2 Hodge theory, which are closely related to intersection ho-mology. Such signatures may relate to duality in string theory, such as throughSen’s conjecture on the dimension of spaces of self-dual harmonic forms onmonopole moduli spaces. Results in these areas and closely related topics in-clude those of Muller [48]; Dai [19]; Cheeger and Dai [17]; Hausel, Hunsicker,and Mazzeo [37; 39; 38]; Saper [51; 50]; Saper and Stern [52];and Carron [13;15; 14]; and work on analytic symmetric signatures is currently being pursuedby Albin, Leichtmann, Mazzeo and Piazza. Much more on analytic approachesto invariants of singular spaces can be found in the other papers in the presentvolume [30].

A different kind of signature invariant that can be defined using non-GMperversities appears in this analytic setting in the works of Hausel, Hunsicker,and Mazzeo [37; 39; 38], in which they demonstrate that groups of L2 har-monic forms on a manifold with fibered boundary can be identified with co-homology spaces associated to the intersection cohomologygroups of varyingperversities for a canonical compactificationX of the manifold. Theseper-verse signaturesare the signatures of the nondegenerate intersection pairings onim.I NpH2n.X

4n/! I NqH2n.X4n; @X 4n//, when Np � Nq. The signature for Witt


spaces mentioned above is a special case in whichNpD NqD NmD Nn and@X D?. IfX is the compactification of the interior of a compact manifoldwith boundary.M; @M / and Np.Z/ < 0 and Nq.Z/ � codim.Z/ � 1 for all singularZ, thenI NpH�.X / Š H�.M /, I NqH�.X / Š H�.M; @M /, and in this case the perversesignature is the classical signature associated to a manifold with boundary.

Using the Lefschetz duality results of general perversity intersection homol-ogy described above, Hunsicker and the author are currentlyundertaking a topo-logical study of the perverse signatures, including research on how Novikovadditivity and Wall non-additivity extend to these settings.

10. Saralegi’s relative intersection chains

Independently of the author’s introduction of stratified coefficients, Saralegi[54] discovered another way, in the case of a constant coefficient system, toobtain an intersection chain complex that satisfies the coneformula (2-1) forgeneral perversities. In [54], he used this chain complex toprove a general per-versity version of the de Rham theorem on unfoldable pseudomanifolds. Thesespaces are a particular type of pseudomanifold on which it ispossible to definea differential form version of intersection cohomology over the real numbers.This de Rham intersection cohomology appeared in a paper by Brylinski [8],though he credits Goresky and MacPherson with the idea. Brylinski showedthat for GM perversities and on a Thom–Mather stratified space, de Rham inter-section cohomology is Hom dual to intersection homology with real coefficients.Working on more general “unfoldable spaces,” Brasselet, Hector, and Saralegilater proved a de Rham theorem in [7], showing that this result can be obtainedby integration of forms on intersection chains, and this wasextended to moregeneral perversities by Saralegi in [53]. However, [53] contains an error in thecase of perversitiesNp satisfying Np.Z/ > codim.Z/� 2 or Np.Z/ < 0 for somesingular stratumZ. This error can be traced directly to the failure of the coneformula for non-GM perversities. Saralegi introduced hisrelative intersectionchains12 in [54] specifically to correct this error.

The rough idea of Saralegi’s relative chains is precisely the same as the au-thor’s motivation for introducing stratified coefficients:when a perversity on astratumZ is too high (greater than codim.Z/�2), it is necessary to kill chainsliving in that stratum in order to preserve the cone formula.The idea of stratifiedcoefficients is to redefine the coefficient system so that suchchains are killedby virtue of their coefficients being trivial. The idea of relative chains is instead

12These should not be confused with relative intersection chains in the senseI NpC�.X;A/Š I NpC�.X /=I

NpC�.A/.

216 GREG FRIEDMAN

to form a quotient group so that the chains living in such strata are killed in thequotient.

More precisely, letA NpCi.X / be the group generated by theNp-allowablei-simplices ofX (notice that there is no requirement that the boundary of anelement ofA NpCi.X / be allowable), and letXNt� Np be the closure of the union ofthe singular strataZ of X such that Np.Z/ > codim.Z/� 2. Let A NpCi.XNt� Np/

be the group generated by theNp allowablei-simplices with support inXNt� Np.Then Saralegi’s relative intersection chain complex is defined to be

S NpC c�.X;XNt� Np/

D

�

A NpC�.X /CA NpC1C�.XNt� Np/�

\@�1�

A NpC��1.X /CA NpC1C��1.XNt� Np/�

A NpC1C�.XNt� Np/\@�1A NpC1C��1.XNt� Np/

:

Roughly speaking, this complex consists ofNp-allowable chains inX andslightly more allowable chains (. NpC 1/-allowable) inXNt� Np whose boundariesare also eitherNp-allowable inX or NpC1 allowable inXNt� Np, but then we quotientout by those chains supported inXNt� Np. This quotient step is akin to the stratifiedcoefficient idea of setting simplices supported inX n�1 to 0. In fact, there is noharm in extending Saralegi’s definition by replacingXNt� Np by all of X n�1, sincethe perversity conditions already guarantee that no simplex of A NpCi.X / northe boundary of any such simplex can have support in those singular strata notin XNt� Np. In addition, there is also nothing special about the choiceNpC 1 forallowability of chains inXNt� Np: the idea is to throw in enough singular chainssupported in the singular strata so that the boundaries of any chains inA NpCi.X /

will also be in “the numerator” (for example, the inallowable0-simplex in@. Ncx/

that lives at the cone vertex in our example in Section 5), butthen to kill anysuch extra chains by taking the quotient. In other words, it would be equivalentto define Saralegi’s relative intersection chain complex as

�

A NpC�.X /CS�.Xn�1/

�

\ @�1�

A NpC��1.X /CS��1.Xn�1/

�

S�.X n�1/;

whereS�.X / is the ordinary singular chain complex.We refer the reader to [28, Appendix A] for a proof13 thatS NpC�.X;XNt� NpIG/

and I NpS�.X IG0/ are chain isomorphic, and so, in particular, they yield thesame intersection homology groups. It is not clear that there is a well-definedversion ofS NpC�.X;XNt� Np/with coefficients in a local systemG defined only on

13The proof in [28] uses a slightly different definition of intersection chains withstratified coefficients than the one given here. However, forany general perversity, theintersection chains with stratified coefficients there are quasi-isomorphic to the ones dis-cussed here, and they are isomorphic for any efficient perversity. See [28, Appendix A].


X�X n�1, and so stratified coefficients may be a slightly broader concept. Theremay also be some technical advantages in sheaf theory to avoiding quotientgroups.

11. Habegger and Saper’s codimension� c intersectionhomology theory

Finally, we discuss briefly the work of Habegger and Saper [35], in whichthey introduce what they callcodimension�c intersection homology. This is thesheafification of King’s loose perversity intersection homology. In a sense, thisis the opposite approach to that of stratified coefficients: stratified coefficientswere introduced to provide a chain theory that agrees with the Deligne sheafconstruction for superperversities, while codimension� c intersection homol-ogy provides a Deligne-type sheaf construction whose hypercohomology yieldsKing’s intersection homology groups. Habegger and Saper work with perversi-ties Np WZ�2!Z such thatNp.k/� Np.kC1/� Np.k/C1 and14 Np.2/�0, and theywork on cs-sets, which generalize pseudomanifolds (see [41; 35]). In fact, Kingshowed in [41] that intersection homology is independent ofthe stratification inthis setting.

The paper [35] involves many technicalities in order to obtain the most gen-eral possible results. We will attempt to simplify the discussion greatly in orderto convey what seems to be the primary stream of ideas. However, we urge thereader to consult [35] for the correct details.

Given a perversityNp, the “codimension� c” in the name of the theory comesfrom considering

c Np Dmin.fk 2 ZCj Np.k/� k � 2g[ f1g/:

In other words,c Np (or simply c when the perversity is understood) is the firstcodimension for whichNp takes the values of a GM perversity. Since the con-dition Np.k/ � Np.k C 1/ � Np.k/C 1 ensures thatNp will be in the Goresky–MacPherson range of values for allk � c, the numberc serves as somewhat of aphase transition. At points in strata of codimension� c, the cone formula (2-1)holds locally for King’s singular intersection chains (i.e., we can use the coneformula to compute the local intersection homology groups in a distinguishedneighborhood). For strata of codimension< c, the perversityNp is in the “super”range, and the cone formula fails, as observed in Section 5. So, the idea ofHabegger and Saper, building on the Goresky–MacPherson–Deligne axiomaticapproach to intersection homology (see Section 4, above) was to find a way toaxiomatize a sheaf construction that upholds the cone formula as the Deligne

14Technically, they allowNp.2/ < 0, but in this case their theory is trivial; see [35, Corollary4.8].

218 GREG FRIEDMAN

sheaf does for GM perversities, but only on strata of codimension� c. Thisidea is successful, though somewhat complicated because the coefficients nowmust live onX � X c and must include the sheafification on this subspace ofU ! I NpS�.U IG/.

In slightly more detail (though still leaving out many technicalities), for afixed Np, let Uc D X � X c Np . Then a codimensionc coefficient systemE� isbasically a sheaf onUc that satisfies the axiomatic properties of the sheafifica-tion of U ! I NpS�.U IG/ there with respect to some stratification ofUc . Theseaxiomatic conditions are a modification of the axiomsAX 2 (see [6, SectionV.4]), which, for a GM perversity, are equivalent to the axiomsAX 1 discussedabove in Section 4. We will not pursue the axiomsAX 2 in detail here, butwe note that the Habegger–Saper modification occurs by requiring certain van-ishing conditions to hold only in certain degrees dependingon c. This takesinto account the failure of the cone formula to vanish in the expected degrees(see Section 5). Then Habegger and Saper define a sheaf complex P�

Np;E� byextendingE� from Uc to the rest ofX by the Deligne process from this point.

Among other results in their paper, Habegger and Saper show that the hy-percohomology of their sheaf complex agrees (up to reindexing and with anappropriate choice of coefficients) with the intersection homology of King onPL pseudomanifolds, that this version of intersection homology is a topologicalinvariant, and that there is a duality theorem. To state their duality theorem,let Nq.k/ D k � 2 � Np.k/, and let Nq0.k/ D max. Nq.k/; 0/ C c Np � 2. Then,with coefficients in a field, the Verdier dualDX P�

Np;E� is quasi-isomorphic toP�

Nq0;DUc .E�/Œcp�2Cn�. Roughly speaking, and ignoring the shifting of perver-

sities and indices, which is done for technical reasons, this says that ifNpC NqD Ntand we dualize the sheaf of intersection chains “by hand” onUc from E� toDUc

E�, then further extensions by the Deligne process, using perversity Np for E�

and perversityNq for DUcE�, will maintain that duality. If Np is a GM perversity

andX is a pseudomanifold with no codimension one strata, this recovers theduality results of Goresky and MacPherson. Unfortunately,for more generalperversities, there does not seem to be an obvious way to translate this dualityback into the language of chain complexes, due to the complexity of the dualcoefficient systemDUc

E� that appears onUc .One additional note should be made concerning the duality results in [35]. As

mentioned above, Habegger and Saper work on cs-sets. These are more generalthan pseudomanifolds, primarily in thatX �X n�1 need not be dense and thereis no inductive assumption that the links be pseudomanifolds. These are thespaces on which King demonstrated his stratification independence results in[41]. Thus these results are more general than those we have been discussingon pseudomanifolds, at least as far as the spaceX is concerned. However, as


far as the author can tell, in one sense these duality resultsare not quite as muchmore general as they at first appear, as least when considering strata ofX thatare not in the closure ofX �X n�1. In particular, ifZ is such a stratum and itlies in Uc , then the duality results on it are tautological — induced bythe “byhand” dualization of the codimensionc coefficient system. But ifZ is not inUc,then the pushforwards of the Deligne process cannot reach it, andP�jZ D 0.So at the sheaf level the truly interesting piece of the duality still occurs in theclosure ofX �X n�1. It would be interesting to understand how the choice ofcoefficient system and “by hand” duality on these “extraneous” strata inUc (thestrata not in the closure ofX �X n�1) influence the hypercohomology groupsand the duality there. We also note that the closure of the union of the regularstrata of a cs-set may still not be a pseudomanifold, due to the lack of conditionon the links. It would be interesting to explore just how muchmore general suchspaces are and the extent to which the other results we have discussed extend tothem.

We refer the reader again to [35] for the further results thatcan be foundthere, including results on the intersection pairing and Zeeman’s filtration.

Acknowledgments

I sincerely thank my co-organizers and co-editors (Eugenie Hunsicker, Ana-toly Libgober, and Laurentiu Maxim) for making an MSRI workshop and thisaccompanying volume possible. I thank my collaborators JimMcClure andEugenie Hunsicker as the impetus and encouragement for some of the work thatis discussed here and for their comments on earlier drafts ofthis paper. AndI thank an anonymous referee for a number of helpful suggestions that havegreatly improved this exposition.

References

[1] Markus Banagl,The signature of partially defined local coefficient systems, J. KnotTheory Ramifications 17 (2008), no. 12, 1455–1481.

[2] , Topological invariants of stratified spaces, Springer Monographs in Math-ematics, Springer, New York, 2006.

[3] Markus Banagl, Sylvain Cappell, and Julius Shaneson,Computing twisted signa-tures and L-classes of stratified spaces, Math. Ann.326(2003), 589–623.

[4] A. A. Beilinson, J. Bernstein, and P. Deligne,Faisceaux pervers, Asterisque100(1982), 5–171.

[5] Armand Borel, Introduction to middle intersection cohomology and perversesheaves, Algebraic groups and their generalizations: classical methods (University

220 GREG FRIEDMAN

Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence,RI, 1994, pp. 25–52.

[6] A. Borel et. al.,Intersection cohomology, Progress in Mathematics, vol. 50, Birk-hauser, Boston, 1984.

[7] J. P. Brasselet, G. Hector, and M. Saralegi,Theeoreme de deRham pour les varietesstratifiees, Ann. Global Anal. Geom.9 (1991), 211–243.

[8] Jean-Luc Brylinski,Equivariant intersection cohomology, Contemp. Math.139(1992), 5–32.

[9] S. E. Cappell, A. Libgober, L. G. Maxim, and J. L. Shaneson, Hodge genera ofalgebraic varieties, II, preprint, see math.AG/0702380.

[10] Sylvain Cappell, Julius Shaneson, and Shmuel Weinberger,Classes topologiquescaracteristiques pour les actions de groupes sur les espaces singuliers, C. R. Acad.Sci. Paris Ser. I Math.313(1991), no. 5, 293–295.

[11] Sylvain E. Cappell, Laurentiu G. Maxim, and Julius L. Shaneson,Hodge generaof algebraic varieties, I, Comm. Pure Appl. Math.61 (2008), no. 3, 422–449.

[12] Sylvain E. Cappell and Julius L. Shaneson,Singular spaces, characteristic classes,and intersection homology, Annals of Mathematics134(1991), 325–374.

[13] G. Carron,L2-cohomology of manifolds with flat ends, Geom. Funct. Anal.13(2003), no. 2, 366–395.

[14] Gilles Carron,CohomologieL2 des varietes QALE, preprint, 2005.

[15] , CohomologieL2 et parabolicite, J. Geom. Anal.15 (2005), no. 3, 391–404.

[16] J. Cheeger,On the Hodge theorey of Riemannian pseudomanifolds, Geometry ofthe Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii,1979) (Providence, RI), vol. 36, Amer. Math. Soc., 1980, pp.91–146.

[17] Jeff Cheeger and Xianzhe Dai,L2-cohomology of spaces with nonisolated conicalsingularities and nonmultiplicativity of the signature. Riemannian topology andgeometric structures on manifolds, 1–24, Progr. Math., 271, Birkhauser, Boston,2009.

[18] Daniel C. Cohen, Mark Goresky, and Lizhen Ji,On the Kunneth formula forintersection cohomology, Trans. Amer. Math. Soc.333(1992), 63–69.

[19] Xianzhe Dai,Adiabatic limits, nonmultiplicativity of signature, and Leray spectralsequence, J. Amer. Math. Soc.4 (1991), no. 2, 265–321.

[20] Alexandru Dimca,Sheaves in topology, Springer, Berlin, 2004.

[21] Greg Friedman,Farber–Levine pairings in intersection homology, in preparation.

[22] , Intersection homology with general perversities, Geom. Dedicata 148(2010), 103–135.

[23] , Stratified fibrations and the intersection homology of the regular neigh-borhoods of bottom strata, Topology Appl.134(2003), 69–109.


[24] , Superperverse intersection cohomology: stratification (in)dependence,Math. Z.252(2006), 49–70.

[25] , Intersection homology of stratified fibrations and neighborhoods, Adv.Math.215(2007), no. 1, 24–65.

[26] , Singular chain intersection homology for traditional and super-perver-sities, Trans. Amer. Math. Soc.359(2007), 1977–2019.

[27] , Intersection homology and Poincare duality on homotopically stratifiedspaces, Geom. Topol.13 (2009), 2163–2204.

[28] Greg Friedman,Intersection homology Kunneth theorems, Math. Ann.343(2009),no. 2, 371–395.

[29] , Intersection homology with field coefficients: K-Witt spaces and K-Wittbordism, Comm. Pure Appl. Math.62 (2009), 1265–1292.

[30] Greg Friedman, Eugenie Hunsicker, Anatoly Libgober, and Laurentiu Maxim,Proceedings of the Workshop on the Topology of Stratified Spaces at MSRI, Sep-tember 8-12, 2008, in preparation.

[31] Greg Friedman and James McClure,Cup and cap product in intersection (co)ho-mology, in preparation.

[32] Mark Goresky and Robert MacPherson,Intersection homology theory, Topology19 (1980), 135–162.

[33] , Intersection homology, II, Invent. Math.72 (1983), 77–129.

[34] Mark Goresky and Paul Siegel,Linking pairings on singular spaces, Comment.Math. Helvetici58 (1983), 96–110.

[35] Nathan Habegger and Leslie Saper,Intersection cohomology of cs-spaces andZeeman’s filtration, Invent. Math.105(1991), 247–272.

[36] Allen Hatcher,Algebraic topology, Cambridge University Press, Cambridge, 2002.

[37] Tamas Hausel, Eugenie Hunsicker, and Rafe Mazzeo,Hodge cohomology ofgravitational instantons, Duke Math. J.122(2004), no. 3, 485–548.

[38] Eugenie Hunsicker,Hodge and signature theorems for a family of manifolds withfibre bundle boundary, Geom. Topol.11 (2007), 1581–1622.

[39] Eugenie Hunsicker and Rafe Mazzeo,Harmonic forms on manifolds with edges,Int. Math. Res. Not. (2005), no. 52, 3229–3272.

[40] Masaki Kashiwara and Pierre Schapira,Sheaves on manifolds, Grundlehren derMathematischen Wissenschaften, vol. 292, Springer, Berlin, 1994.

[41] Henry C. King,Topological invariance of intersection homology without sheaves,Topology Appl.20 (1985), 149–160.

[42] Frances Kirwan and Jonathan Woolf,An introduction to intersection homologytheory. second edition, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[43] Steven L. Kleiman,The development of intersection homology theory, Pure Appl.Math. Q.3 (2007), 225–282.

222 GREG FRIEDMAN

[44] Robert MacPherson,Intersection homology and perverse sheaves, UnpublishedColloquium Lectures.

[45] R. MacPherson and K. Vilonen,Elementary construction of perverse sheaves,Invent. Math.84 (1986), 403–435.

[46] J. E. McClure,On the chain-level intersection pairing for PL manifolds, Geom.Topol.10 (2006), 1391–1424.

[47] Clint McCrory, Stratified general position, Algebraic and geometric topology(Proc. Sympos., Univ. California, Santa Barbara, Calif. 1977) (Berlin), Lecture Notesin Math., vol. 664, Springer, 1978, pp. 142–146.

[48] Werner Muller, L2-index theory, eta invariants and values ofL-functions, Geo-metric and topological invariants of elliptic operators (Brunswick, ME, 1988), Con-temp. Math., vol. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 145–189.

[49] James R. Munkres,Elements of algebraic topology, Addison-Wesley, Reading,MA, 1984.

[50] Leslie Saper,L2-cohomology of locally symmetric spaces, I, Pure Appl. Math. Q.1 (2005), no. 4, part 3, 889–937.

[51] , L-modules and the conjecture of Rapoport and Goresky-MacPherson,Automorphic forms. I. Asterisque (2005), no. 298, 319–334.

[52] Leslie Saper and Mark Stern,L2-cohomology of arithmetic varieties, Ann. ofMath. (2)132(1990), no. 1, 1–69.

[53] Martin Saralegi,Homological properties of stratified spaces, I, Illinois Journal ofMathematics38 (1994), 47–70.

[54] Martintxo Saralegi-Aranguren,de Rham intersection cohomology for generalperversities, Illinois J. Math.49 (2005), no. 3, 737–758.

[55] P. H. Siegel,Witt spaces: a geometric cycle theory for KO-homology at oddprimes,American J. Math.110(1934), 571–92.

[56] Norman Steenrod,The topology of fibre bundles, Princeton University Press,Princeton, NJ, 1951.

GREG FRIEDMAN


TEXAS CHRISTIAN UNIVERSITY

BOX 298900FORT WORTH, TX 76129

[email protected]


The signature of singular spaces and itsrefinements to generalized homology theories

MARKUS BANAGL

ABSTRACT. These notes are based on an expository lecture that I gave attheworkshop “Topology of Stratified Spaces” at MSRI Berkeley inSeptember2008. We will first explain the definition of a bordism invariant signature for asingular space, proceeding along a progression from less singular to more andmore singular spaces, starting out from spaces that have no odd codimensionalstrata and, after having discussed Goresky–Siegel spaces and Witt spaces, end-ing up with general (non-Witt) stratified spaces. We will moreover discussvarious refinements of the signature to orientation classesin suitable bordismtheories based on singular cycles. For instance, we will indicate how one maydefine a symmetricL�-homology orientation for Goresky–Siegel spaces or aSullivan orientation for those non-Witt spaces that still possess generalizedPoincare duality. These classes can be thought of as refining the L-class ofa singular space. Along the way, we will also see how to compute twistedversions of the signature and L-class.

CONTENTS

1. Introduction 2242. Pseudomanifolds without odd codimensional strata 2273. Witt spaces 2294. IP spaces: integral duality 2325. Non-Witt spaces 236References 247

The author was in part supported by a research grant of the Deutsche Forschungsgemeinschaft. He thanksAndrew Ranicki for suggestions and clarifying comments.

223

224 MARKUS BANAGL

1. Introduction

Let M be a closed smoothn-dimensional manifold. The Hirzebruch L-classesLi.M / 2 H 4i.M IQ/ of its tangent bundle are powerful tools in theclassification of suchM , particularly in the high dimensional situation wheren� 5. To make this plausible, we observe first that theLi.M /, with the excep-tion of the top classLn=4.M / if n is divisible by4, are not generally homotopyinvariants ofM , and are therefore capable of distinguishing manifolds in agivenhomotopy type, contrary to the ability of homology and otherhomotopy invari-ants. For example, there exist infinitely many manifoldsMi ; i D 1; 2; : : : in thehomotopy type ofS2 � S4, distinguished by the first Pontrjagin class of theirtangent bundlep1.TMi/ 2 H 4.S2 � S4/ D Z, namelyp1.TMi/ D Ki , K afixed nonzero integer. The first L-classL1 is proportional to the first Pontrjaginclassp1, in fact they are related by the formulaL1 D 1

3p1.

Suppose thatM n, n� 5, is simply connected, as in the example. The classi-fication of manifolds breaks up into two very different tasks: Classify Poincarecomplexes up to homotopy equivalence and, given a Poincare complex, deter-mine all manifolds homotopy equivalent to it.

In dimension3, one has a relatively complete answer to the former problem.One can associate purely algebraic data to a Poincare complex such that twosuch complexes are homotopy equivalent if, and only if, their algebraic data areisomorphic, see the classification result in [Hen77]. Furthermore, every givenalgebraic data is realizable as the data of a Poincare complex; see [Tur90]. Inhigher dimensions, the problem becomes harder. While one can still associateclassifying data to a Poincare complex, this data is not purely algebraic anymore,though at least in dimension4, one can endow Poincare duality chain complexeswith an additional structure that allows for classification, [BB08].

The latter problem is the realm of surgery theory. Elements of the structuresetS.M / of M are represented by homotopy equivalencesN !M , whereN

is another closed smooth manifold, necessarily simply connected, sinceM is.Two such homotopy equivalences represent the same element of S.M / if thereis a diffeomorphism between the domains that commutes with the homotopyequivalences. The goal of surgery theory is to computeS.M /. The central toolprovided by the theory is the surgery exact sequence

LnC1� S.M /��N.M /�Ln;

an exact sequence of pointed sets. TheLn are the4-periodic simply connectedsurgery obstruction groups,Ln D Z; 0;Z=2; 0 for n � 0; 1; 2; 3 mod 4. Theterm N.M / is thenormal invariant set, investigated by Sullivan. It is a gen-eralized cohomology theory and a Pontrjagin–Thom type construction yieldsN.M /Š ŒM;G=O �, whereŒM;G=O � denotes homotopy classes of maps from

THE SIGNATURE OF SINGULAR SPACES AND ITS REFINEMENTS 225

M into a certain universal spaceG=O, which does not depend onM . SinceŒM;G=O � is a cohomology theory, it is particularly important to knowits coef-ficients��.G=O/. While the torsion is complicated, one has modulo torsion

�i.G=O/˝QD

�

Q; i D 4j ,0; otherwise.

One obtains an isomorphism

ŒM;G=O �˝QŠM

j�0

H 4j .M IQ/:

The groupLnC1 acts onS.M / so that the point-inverses of� are the orbits ofthe action, i.e. for allf; h 2 S.M / one has�.f /D �.h/ if, and only if, there isa g 2LnC1 which movesf to h, g �f D h.

Suppose our manifoldM is even dimensional. ThenLnC1 vanishes and thus�.f / D �.h/ implies f D g � f D h, so that� is injective. In particular, weobtain an injection

S.M /˝QŒN.M /˝Q:

Composing this withN.M /˝QŠL

H 4j .M IQ/, we obtain an injective map

S.M /˝QLŒ

L

H 4j .M IQ/:

This map sends a homotopy equivalenceh W N !M to the cohomology classL�.h/ uniquely determined byh�.L�.M /C L�.h// D L�.N /. ThusM isdetermined, up to finite ambiguity, byits homotopy type and its L-classes. Thisdemonstrates impressively the power of the L-classes as a tool to classify man-ifolds.

The L-classes are closely related to the signature invariant, and indeed theclasses can be defined, following Thom [Tho58], by the signatures of submani-folds, as we shall now outline. The link between the L-classes and the signatureis the Hirzebruch signature theorem. It asserts that the evaluation of the topL-classLj .M / 2 H n.M IQ/ of annD 4j -dimensional oriented manifoldMon the fundamental class ofM equals the signature�.M / of M . Once weknow this, we can defineL�.M / as follows. A theorem of Serre states that theHurewicz map is an isomorphism

�k.M /˝QŠH k.M IQ/

in the rangen<2k�1, where��.M / denotes the cohomotopy sets ofM , whoseelements are homotopy classes of maps fromM to spheres. Thus, in this range,we may think of a rational cohomology class as a (smooth) mapf WM ! Sk .The preimagef �1.p/ of a regular valuep 2Sk is a submanifold and has a sig-nature�.f �1.p//. Use the bordism invariance of the signature to conclude that

226 MARKUS BANAGL

this signature depends only on the homotopy class off . Assigning�.f �1.p//

to the homotopy class off yields a mapH k.M IQ/! Q, that is, ahomologyclassLk.M /2Hk.M IQ/. By Poincare duality, this class can be dualized backinto cohomology, where it agrees with the Hirzebruch classes L�.M /. Notethat all you need for this procedure is transversality for maps to spheres in orderto get suitable subspaces and a bordism invariant signaturedefined on thesesubspaces. Thus, whenever these ingredients are present for a singular spaceX , we will obtain an L-classL�.X / 2 H�.X IQ/ in the rational homology ofX . (This class cannot necessarily be dualized back into cohomology, due tothe lack of classical Poincare duality for singularX .) Therefore, we only needto discuss which classes of singular spaces have a bordism invariant signature.The required transversality results are available for Whitney stratified spaces,for example. The notion of a Whitney stratified space incorporates smoothnessin a particularly amenable way into the world of singular spaces. A Whitneystratification of a spaceX consists of a (locally finite) partition ofX into locallyclosed smooth manifolds of various dimensions, called thepure strata. If onestratum intersects the closure of another one, then it must already be completelycontained in it. Connected componentsS of strata have tubular neighborhoodsTS that possess locally trivial projections�S W TS ! S whose fiber��1

S.p/,

p 2 S , is the cone on a compact spaceL.p/ (also Whitney stratified), calledthe link of S at p. It follows that every pointp has a neighborhood homeo-morphic toRdimS �coneL.p/. Real and complex algebraic varieties possess anatural Whitney stratification, as do orbit spaces of smoothgroup actions. Thepseudomanifold condition means that the singular strata have codimension atleast two and the complement of the singular set (thetop stratum) is dense inX . The figure eight space, for instance, can be Whitney stratified but is not apseudomanifold. The pinched2-torus is a Whitney stratifiable pseudomanifold.If we attach a whisker to the pinched2-torus, then it loses its pseudomanifoldproperty, while retaining its Whitney stratifiability. By [Gor78], a Whitney strat-ified pseudomanifoldX can be triangulated so that the Whitney stratificationdefines a PL stratification ofX .

Inspired by the success of L-classes in manifold theory sketched above, onewould like to have L-classes for stratified pseudomanifoldsas well. In [CW91],see also [Wei94], Cappell and Weinberger indicate the following result, anal-ogous to the manifold classification result sketched above.SupposeX is astratified pseudomanifold that has no strata of odd dimension. Assume that allstrataS have dimension at least5, and that all fundamental groups in sight aretrivial, that is, all strata are simply connected and all links are simply connected.(A pseudomanifold whose links are all simply connected is called supernormal.This is compatible with the notion of anormal pseudomanifold, meaning that


all links are connected.) Then differences of L-classes give an injection

S.X /˝QŒM

S�X

M

j

Hj .S IQ/;

whereS ranges over the strata ofX , S denotes the closure ofS in X , andS.X / is an appropriately1 defined structure set forX . This would suggest thatL-classes are as powerful in classifying stratified spaces as in classifying man-ifolds. Since, as we have seen, the definition of L-classes isintimately relatedto, and can be given in terms of, the signature, we shall primarily investigatethe possibility of defining a bordism invariant signature for an oriented stratifiedpseudomanifoldX .

2. Pseudomanifolds without odd codimensional strata

In order to define a signature, one needs an intersection form. But singularspaces do not possess Poincare duality, in particular no intersection form, inordinary homology. The solution is to change to a different kind of homology.Motivated by a question of D. Sullivan [Sul70], Goresky and MacPherson define(in [GM80] for PL pseudomanifolds and in [GM83] for topological pseudoman-ifolds) a collection of groupsIH

Np� .X /, called intersection homology groups

of X , depending on a multi-indexNp, called aperversity. For these groups, aPoincare–Lefschetz-type intersection theory can be defined, and a generalizedform of Poincare duality holds, but only between groups with “complementaryperversities.” More precisely, withNt.k/ D k � 2 denoting the top perversity,there are intersection pairings

IHNp

i .X /˝ IHNq

j .X /� Z (2-1)

for an oriented closed pseudomanifoldX , NpC NqD Nt andiCj D dimX , whichare nondegenerate when tensored with the rationals. Jeff Cheeger discovered,working independently of Goresky and MacPherson and not being aware of theirintersection homology, that Poincare duality on triangulated pseudomanifoldsequipped with a suitable (locally conical) Riemannian metric on the top stra-tum, can be recovered by using the complex ofL2 differential forms on the topstratum, see [Che80], [Che79] and [Che83]. The connection between his and thework of Goresky and MacPherson was pointed out by Sullivan in1976. For anintroduction to intersection homology see [BC84], [KW06] or [Ban07]. A thirdmethod, introduced in [Ban09] and implemented there for pseudomanifolds

1In [CW91], the structure setsS.X / are defined as the homotopy groups of the homotopy fiber of theassembly mapX ^ L�.Z/0 ! L�.Z�1.X //, constructed in [Ran79]. This can be defined for any space,but under the stated assumptions onX , [CW91] interpretsS.X / geometrically in terms of classical structuresets of the strata ofX .

228 MARKUS BANAGL

with isolated singularities and two-strata spaces with untwisted link bundle,associates to a singular pseudomanifoldX an intersection spaceI NpX , whoseordinary rational homology has a nondegenerate intersection pairing

eH i.INpX IQ/˝ eHj .I

NqX IQ/�Q:

This theory is not isomorphic, albeit related, to intersection homology. It solvesa problem posed in string theory, related to the presence of massless D-branesin the course of conifold transitions.

In sheaf-theoretic language, the groupsIHNp

� .X / are given as the hypercoho-mology groups of a sheaf complexIC �

Np.X / over X . If we view this complexas an object of the derived category (that is, we invert quasi-isomorphisms),thenIC �

Np.X / is characterized by certain stalk/costalk vanishing conditions. Therationalization of the above intersection pairing (2-1) isinduced on hypercoho-mology by a duality isomorphismDIC �

Np.X IQ/Œn�Š IC �

Nq.X IQ/ in the derivedcategory, whereD denotes the Verdier dualizing functor. This means roughlythat one does not just have a global chain equivalence to the dual (intersection)chain complex, but a chain equivalence on every open set.

Let X n be an oriented closed topological stratified pseudomanifold whichhas only even dimensional strata. A wide class of examples isgiven by complexalgebraic varieties. In this case, the intersection pairing (2-1) allows us to definea signature�.X / by using the two complementary middle perversitiesNm and Nn:

k 2 3 4 5 6 7 8 9 : : :

Nm.k/ 0 0 1 1 2 2 3 3 : : :

Nn.k/ 0 1 1 2 2 3 3 4 : : :

Since Nm.k/D Nn.k/ for even values ofk, and only these values are relevant forour presentX , we haveIH Nm

n=2.X / D IH Nn

n=2.X /. Therefore, the pairing (2-1)

becomes

IH Nmn=2.X IQ/˝ IH Nm

n=2.X IQ/�Q

(symmetric ifn=2 is even), that is, defines a quadratic form on the vector spaceIH Nm

n=2.X IQ/. Let �.X / be the signature of this quadratic form. Goresky and

MacPherson show that this is a bordism invariant for bordisms that have onlystrata of even codimension. SinceIC �

Nm.X /D IC �

Nn.X /, the intersection pairingis induced by a self-duality isomorphismDIC �

Nm.X IQ/Œn�Š IC �

Nm.X IQ/. Thisis an example of aself-dual sheaf.


3. Witt spaces

To form a bordism theory based on pseudomanifold cycles, onecould con-sider bordism based on all (say topological, or PL) closed pseudomanifolds,

˝all pseudomfds� .Y /D

˚

ŒXf� Y � j X a pseudomanifold

;

where the admissible bordisms consist of compact pseudomanifolds with col-lared boundary, without further restrictions. Now it is immediately clear thatthe associated coefficient groups vanish,˝

all pseudomfds� .pt/D 0, �> 0, since any

pseudomanifoldX is the boundary of the cone onX , which is an admissiblebordism. Thus this naive definition does not lead to an interesting and usefulnew theory, and we conclude that a subclass of pseudomanifolds has to be se-lected to define such theories. Given the results on middle perversity intersectionhomology presented so far, our next approach would be to select the class of allclosed pseudomanifolds with only even codimensional strata,

˝ev�.Y /D

˚

ŒXf� Y � j X has only even codim strata

(and the same condition is imposed on all admissible bordisms). While wedo know that the signature is well-defined onev

�.pt/, this is however still

not a good theory as this definition leads to a large number of geometricallyinsignificant generators. Many operations (such as coning or refinement of thestratification) do introduce strata of odd codimension, so we need to allow somestrata of this kind, but so as not to destroy Poincare duality. In [Sie83], PaulSiegel introduced a class of oriented stratified PL pseudomanifolds calledWittspaces, by imposing the condition thatIH Nm

middle.Link.x/IQ/D 0 for all pointsx in odd codimensional strata ofX . The suspensionX 7 D ˙CP

3 has twosingular points which form a stratum of odd codimension7. The link is CP

3

with middle homologyH3.CP3/D0. HenceX 7 is a Witt space. The suspension

X 3D˙T 2 has two singular points which form a stratum of odd codimension 3.The spaceX 3 is not Witt, since the middle Betti number of the linkT 2 is 2. Insheaf-theoretic language, a pseudomanifoldX is Witt if and only if the canon-ical morphismIC �

Nm.X IQ/ ! IC �

Nn.X IQ/ is an isomorphism (in the derivedcategory). Thus,IC �

Nm.X IQ/ is self-dual on a Witt space, and ifX is compact,we have a nonsingular intersection pairingIH Nm

i .X IQ/˝ IH Nmn�i.X IQ/! Q.

Let ˝Witt�.Y / denote Witt space bordism, that is, bordism of closed oriented

Witt spacesX mapping continuously intoY . Admissible bordisms are compactpseudomanifolds with collared boundary that satisfy the Witt condition, togetherwith a map intoY .

When is a Witt spaceX n a boundary? Suppose the dimensionn is odd. ThenX D @Y with Y D coneX . The coneY is a Witt space, since the cone-point is

230 MARKUS BANAGL

a stratum of even codimension inY . This shows that Witt2kC1

.pt/D 0 for all k.In particular, the de Rham invariant does not survive in˝Witt

�.pt/.

Let W .Q/ denote the Witt group of the rationals. Its structure is known andgiven by

W .Q/ŠW .Z/˚M

p prime

W .Z=p/;

whereW .Z/Š Z via the signature,W .Z=2/Š Z=2, and forp 6D 2, W .Z=p/Š

Z=4 or Z=2 ˚ Z=2. Sending a Witt spaceX 4k to its intersection form onIH Nm

2k.X IQ/ defines a bordism-invariant elementw.X / 2W .Q/. Siegel shows

that the induced mapw W ˝Witt4k.pt/! W .Q/ is an isomorphism fork > 0. In

dimension zero we get Witt0D Z. If X has dimension congruent2 modulo4,

thenX bounds a Witt space by singular surgery on a symplectic basisfor theantisymmetric intersection form. ThusWitt

n D 0 for n not congruent0 mod4.SinceW .Q/ is just another name for the L-groupL4k.Q/ andLn.Q/D 0 for n

not a multiple of4, we can summarize Siegel’s result succinctly as saying that˝Witt

� .pt/ŠL�.Q/ in positive degrees. By the Brown representability theorem,Witt space bordism theory is given by a spectrum MWITT, whichis in fact anMSO module spectrum, see [Cur92]. (Regard a manifold as a Witt space withone stratum.) By [TW79], any MSO module spectrum becomes a product ofEilenberg–Mac Lane spectra after localizing at2. Thus,

MWITT .2/ 'K.Z.2/; 0/�Y

j>0

K.Lj .Q/.2/; j /

and we conclude that

˝Wittn .Y /.2/ ŠHn.Y IZ/.2/˚

M

j>0

Hn�j .Y ILj .Q/.2//:

(As Z.2/ is flat overZ, we haveS�.X /.2/ D .S.2//�.X / for any spectrumS .)Let us focus on the odd-primary situation. RegardZŒ1

2; t � as a graded ring with

deg.t/D 4. Let˝SO�.Y / denote bordism of smooth oriented manifolds. Consid-

ering the signature as a map� W˝SO�.pt/! ZŒ1

2; t �, ŒM 4k �‘ �.M /tk , makes

ZŒ12; t � into an˝SO

�.pt/-module and we can form the homology theory

˝SO�.Y /˝˝SO

� .pt/ ZŒ12; t �:

On a point, this is

˝SO�.pt/˝˝SO

� .pt/ ZŒ12; t �Š ZŒ1

2; t �;

the isomorphism being given byŒM 4l �˝atk‘ a�.M 4l/tkCl . Let ko�.Y / de-note connective KO homology, regarded as aZ-graded, notZ=4-graded, theory.


It is given by a spectrumbo whose homotopy groups vanish in negative degreesand are given by

ko�.pt/D ��.bo/D Z˚˙1Z=2˚˙

2Z=2˚˙

4Z˚˙8

Z˚˙9Z=2˚ � � � ;

repeating with8-fold periodicity in nonnegative degrees. Inverting2 kills thetorsion in degrees1 and2 mod8 so thatko�.pt/˝Z ZŒ1

2�ŠZŒ1

2; t �. In his MIT-

notes [Sul05], Sullivan constructs a natural Conner–Floyd-type isomorphism ofhomology theories

˝SO�.Y /˝˝SO

� .pt/ ZŒ12; t �

Š

� ko�.Y /˝Z ZŒ12�:

Siegel [Sie83] shows that Witt spaces provide a geometric description of con-nective KO homology at odd primes: He constructs a natural isomorphism ofhomology theories

˝Witt�.Y /˝Z ZŒ1

2�

Š

� ko�.Y /˝Z ZŒ12� (3-1)

(which we shall return to later). It reduces to the signaturehomomorphism oncoefficients, i.e. an elementŒX 4k �˝a 2˝Witt

4k.pt/˝Z ZŒ1

2� maps toa�.X /tk 2

ko�.pt/˝Z ZŒ12� D ZŒ1

2; t �. This is an isomorphism, since inverting2 kills the

torsion components of the invariantw.X /, W .Q/˝ZŒ12�ŠW .Z/˝ZŒ1

2�ŠZŒ1

2�.

Now˝SO� .Y /˝˝SO

� .pt/ ZŒ12; t � being a quotient of SO

� .Y /˝Z ZŒ12; t �, yields a

natural surjection

˝SO� .Y /˝Z ZŒ1

2; t �“˝SO

� .Y /˝˝SO� .pt/ ZŒ1

2; t �:

Let us consider the diagram of natural transformations

˝SO� .Y /˝˝SO

� .pt/ ZŒ12; t �

Š// ko�.Y /˝Z ZŒ1

2�

˝SO�.Y /˝Z ZŒ1

2; t �

OO

OO

// ˝Witt�.Y /˝Z ZŒ1

2�;

Š

OO

where the lower horizontal arrow maps an elementŒMf� Y �˝ atk to ŒM �

CP2kf �1

� Y �˝ a. On a point, this arrow thus maps an elementŒM 4l �˝ atk

to ŒM 4l � CP2k � ˝ a. Mapping an elementŒM 4l � ˝ atk clockwise yieldsa�.M /tkCl 2 ko�.pt/˝Z ZŒ1

2� Š ZŒ1

2; t �. Mapping the same element coun-

terclockwise gives

a�.M 4l �CP2k/t .4lC4k/=4 D a�.M /tkCl

also. The diagram commutes and shows that away from2, the canonical mapfrom manifold bordism to Witt bordism is a surjection. This is a key observationof [BCS03] and frequently allows bordism invariant calculations for Witt spaces

232 MARKUS BANAGL

to be pulled back to calculations on smooth manifolds. This principle may beviewed as a topological counterpart of resolution of singularities in complexalgebraic and analytic geometry (though one should point out that there are com-plex 2-dimensional singular projective toric varietiesX.�/ such that no nonzeromultiple of X.�/ is bordant to any toric resolution ofX.�/). It is applied in[BCS03] to prove that the twisted signature�.X I S/ of a closed oriented Whit-ney stratified Witt spaceX n of even dimension with coefficients in a (Poincare-)local systemS on X can be computed as the product of the (untwisted) L-classof X and a modified Chern character of the K-theory signatureŒS�K of S,

�.X I S/D hechŒS�K ;L�.X /i 2 Z

whereL�.X / 2 H�.X IQ/ is the total L-class ofX . The higher componentsof the productechŒS�K \L�.X / in fact compute the rest of the twisted L-classL�.X I S/. Such twisted classes come up naturally if one wants to understandthe pushforward under a stratified map of characteristic classes of the domain,see [CS91] and [Ban06c].

4. IP spaces: integral duality

Witt spaces satisfy generalized Poincare duality rationally. Is there a classof pseudomanifolds whose members satisfy Poincare duality integrally? Thisrequires restrictions more severe than those imposed on Witt spaces. Anin-tersection homology Poincare space(“IP space”), introduced in [GS83], is anoriented stratified PL pseudomanifold such that the middle perversity, middledimensional intersection homology of even dimensional links vanishes and thetorsion subgroup of the middle perversity, lower middle dimensional intersec-tion homology of odd dimensional links vanishes. This condition characterizesspaces for which the integral intersection chain sheafIC �

Nm.X IZ/ is self-dual.Goresky and Siegel show that for such spacesX n there are nonsingular pairings

IH Nmi .X /=Tors˝ IH Nm

n�i.X /=Tors� Z

and

TorsIH Nmi .X /˝TorsIH Nm

n�i�1.X /�Q=Z:

Let˝ IP�.pt/ denote the bordism groups of IP spaces. The signature�.X / of the

above intersection pairing is a bordism invariant and induces a homomorphism˝ IP

4k.pt/!Z. If dim X DnD4kC1, then the number mod2 of Z=2-summands

in TorsIH Nm2k.X / is a bordism invariant, thede Rham invariantdR.X / 2 Z=2

of X . It induces a homomorphism dRW ˝ IP4kC1

.pt/! Z=2. Pardon shows in[Par90] that these maps are both isomorphisms fork�1 and that all other groups


˝ IPn .pt/D 0, n> 0. In summary, one obtains

˝ IPn .pt/D

(

Z; n� 0.4/,Z=2; n� 5; n� 1.4/,0 otherwise.

LetL�.ZG/ denote the symmetric L-groups, as defined by Ranicki, of the groupring ZG of a groupG. For the trivial groupG D e, these are the homotopygroups��.L

�/D L�

�.pt/ of the symmetric L-spectrumL� and are given by

Ln.Ze/D

(

Z; n� 0.4/,Z=2; n� 1.4/,0 otherwise.

We notice that this is extremely close to the IP bordism groups, the only differ-ence being aZ=2 in dimension1. A comparison of their respective coefficientgroups thus leads us to expect that the difference between the generalized homol-ogy theory˝ IP

� .Y / given by mapping IP pseudomanifoldsX continuously intoa spaceY and symmetricL�-homologyL

�

�.Y / is very small. Indeed, accordingto [Epp07], there exists a map� WMIP!L

�, where MIP is the spectrum givingrise to IP bordism theory, whose homotopy cofiber is an Eilenberg–Mac LanespectrumK.Z=2; 1/. The map is obtained by using a description ofL

� as asimplicial˝-spectrum, whosek-th space has itsn-simplices given by homotopyclasses of.n�k/-dimensionaln-ads of symmetric algebraic Poincare complexes(pairs). Similarly, MIP can be described as a simplicial˝-spectrum, whosek-th space has itsn-simplices given by.n� k/-dimensionaln-ads of compact IPpseudomanifolds. Given these simplicial models, one has tomapn-ads of IPspaces ton-ads of symmetric Poincare complexes. On a suitable incarnation ofthe middle perversity integral intersection chain sheaf ona compact IP space,a Poincare symmetric structure can be constructed by copying Goresky’s sym-metric construction of [Gor84]. Taking global sections andresolving by finitelygenerated projectives (observing that the cohomology of the section complex isfinitely generated by compactness), one obtains a symmetricalgebraic Poincarecomplex. This assignment can also be done for pairs and behaves well undergluing. The symmetric structure is uniquely determined by its restriction to thetop stratum. On the top stratum, which is a manifold, the construction agreessheaf-theoretically with the construction used classically for manifolds, see e.g.[Bre97]. In particular, if we start with a smooth oriented closed manifold andview it as an IP space with one stratum, then the top stratum istheentirespaceand the constructed symmetric structure agrees with Ranicki’s symmetric struc-ture. Modelling MSO and MSTOP as simplicial -spectra consisting ofn-ads of smooth oriented manifolds andn-ads of topological oriented manifolds,

234 MARKUS BANAGL

respectively, we thus see that the diagram

MSO //

››

MIP�

››

MSTOP //

L�

homotopy commutes, where the “symmetric signature map” MSTOP! L� of

ring spectra has been constructed by Ranicki. (Technically, IP spaces are PLpseudomanifolds, so to obtain the canonical map MSO!MIP, it is necessaryto find a canonical PL structure on a given smooth manifold. This is possibleby J. H. C. Whitehead’s triangulation results of [Whi40], where it is shown thatevery smooth manifold admits a compatible triangulation asa PL manifold andthis PL manifold is unique to within a PL homeomorphism; see also [WJ66].)It follows that

˝SO�.Y / //

››

˝ IP�.Y /

��.Y /››

˝STOP� .Y / // L

�

�.Y /

(4-1)

commutes. Let us verify the commutativity forY D pt by hand. If� D 4kC 2

or 4kC3, thenL�

�.pt/DL�.Ze/D 0, so the two transformations agree in these

dimensions. Commutativity in dimension1 follows from ˝SO1.pt/ D 0. For

� D 4kC 1, k > 0, the homotopy cofiber sequence of spectra

MIP�� L

��K.Z=2; 1/

induces on homotopy groups an exact sequence and hence an isomorphism

�4kC1.MIP/Š

� �4kC1.L�/:

But both of these groups areZ=2, whence the isomorphism is the identity map.Thus ifM 4kC1 is a smooth oriented manifold, thenŒM 4kC1�2˝ IP

4kC1.pt/maps

under� to the de Rham invariant dR.M / 2L4kC1.Ze/D Z=2. Hence the twotransformations agree on a point in dimensions4k C 1. Again using the exactsequence of homotopy groups determined by the above cofibration sequence,

� induces isomorphisms�4k.MIP/Š

� �4k.L�/. Both of these groups areZ,

so this isomorphism is 1. Consequently, a smooth oriented manifoldM 4k ,defining an elementŒM 4k �2˝ IP

4k.pt/, maps under� to˙�.M /2L4k.Ze/DZ,

and it isC�.M / when the signs in the two symmetric structures are correctlymatched.

For an n-dimensional Poincare space which is either a topological mani-fold or a combinatorial homology manifold (i.e. a polyhedron whose links of


simplices are homology spheres), Ranicki defines a canonical L�-orientation

ŒM �L 2 L�

n.M /, see [Ran92]. Its image under the assembly map

L�

n.M /A�Ln.Z�1.M //

is thesymmetric signature��.M /, which is a homotopy invariant. The classŒM �L itself is a topological invariant. The geometric meaning oftheL

�-orienta-tion class is that its existence for a geometric Poincare complexX n, n � 5,assembling to the symmetric signature (which any Poincare complex possesses),implies up to 2-torsion thatX is homotopy equivalent to a compact topologicalmanifold. (More precisely,X is homotopy equivalent to a compact manifold if ithas anL�-orientation class, which assembles to thevisiblesymmetric signatureof X .) Cap product withŒM �L induces anL�-homology Poincare duality iso-

morphism.L�/i.M /Š

�L�

n�i.M /. Rationally,ŒM �L is given by the homologyL-class ofM ,

ŒM �L˝ 1DL�.M / 2 L�

n.M /˝QŠM

j�0

Hn�4j .M IQ/:

Thus, we may viewŒM �L as an integral refinement of the L-class ofM . An-other integral refinement of the L-class is the signature homology orientationclassŒM �Sig 2 Sign.M /, to be defined below. The identityAŒM �L D �

�.M /

may then be interpreted as a non-simply connected generalization of the Hirze-bruch signature formula. The localization ofŒM �L at odd primes is the Sul-livan orientation�.M / 2 KOn.M / ˝ ZŒ1

2�, which we shall return to later.

Under the map STOPn .M /! L

�

n.M /, ŒM �L is the image of the identity map

ŒMid�M � 2˝STOP

n .M /.

We shall now apply� in defining anL�-orientationŒX �L 2 L

�

n.X / for anoriented closedn-dimensional IP pseudomanifoldX . (For Witt spaces, anL�-orientation and a symmetric signature has been defined in [CSW91].) The iden-tity mapX !X defines an orientation classŒX �IP 2˝ IP

n .X /.

DEFINITION 4.1. TheL�-orientationŒX �L 2 L

�

n.X / of an oriented closedn-dimensional IP pseudomanifoldX is defined to be the image ofŒX �IP2˝ IP

n .X /

under the map

˝ IPn .X /

��.X /� L

�

n.X /:

If X D M n is a smooth oriented manifold, then the identity mapM ! M

defines an orientation classŒM �SO2 ˝SOn .M /, which maps toŒM �L under the

map

˝SOn .M /�˝STOP

n .M /� L�

n.M /:

236 MARKUS BANAGL

Thus, the above definition ofŒX �L for an IP pseudomanifoldX is compatiblewith manifold theory in view of the commutativity of diagram(4-1). Apply-ing Ranicki’s assembly map, it is then straightforward to define the symmetricsignature of an IP pseudomanifold.

DEFINITION 4.2. The symmetric signature��.X / 2 Ln.Z�1.X // of an ori-ented closedn-dimensional IP pseudomanifoldX is defined to be the image ofŒX �L under the assembly map

L�

n.X /A�Ln.Z�1.X //:

This then agrees with the definition of the Mishchenko–Ranicki symmetric sig-nature��.M / of a manifoldX DM becauseAŒM �L D �

�.M /.

5. Non-Witt spaces

All pseudomanifolds previously considered had to satisfy avanishing con-dition for the middle dimensional intersection homology ofthe links of oddcodimensional strata. Can a bordism invariant signature bedefined for an evenlarger class of spaces? As pointed out above, taking the coneon a pseudoman-ifold immediately proves the futility of such an attempt on the full class of allpseudomanifolds. What, then, are the obstructions for an oriented pseudoman-ifold to possess Poincare duality compatible with intersection homology?

Let LK be a collection of closed oriented pseudomanifolds. We might envi-sion forming a bordism group LK

n , whose elements are represented by closedorientedn-dimensional stratified pseudomanifolds whose links are all home-omorphic to (finite disjoint unions of) elements ofLK. Two spacesX andX 0 represent the same bordism class,ŒX � D ŒX 0�, if there exists an.nC 1/-dimensional oriented compact pseudomanifold-with-boundary Y nC1 such thatall links of the interior ofY are inLK and@Y ŠX t�X 0 under an orientation-preserving homeomorphism. (The boundary is, as always, to be collared in astratum-preserving way.) If, for instance,

LKD fS1;S2;S3; : : :g;

then˝LK�

is bordism of manifolds. If

LKDOdd[fL2l j IH Nml .LIQ/D 0g;

where Odd is the collection of all odd dimensional oriented closed pseudoman-ifolds, then˝LK

� D˝Witt� . The question is: Which other spaces can one throw

into thisLK, yielding an enlarged collectionLK0 �LK, such that one can still


define a bordism invariant signature� W˝LK0

�� Z so that the diagram

˝Witt�

�//

››

Z

˝LK0

�

�

<<

commutes, where Witt��˝LK0

�is the canonical map induced by the inclusion

LK� LK0? Note that�.L/D 0 for everyL 2 LK. Suppose we took anLK

0

that contains a manifold P with�.P / 6D0, e.g.PDCP2k . ThenŒP �D02˝LK0

�,

sinceP is the boundary of the cone onP , and the cone onP is an admissiblebordism in˝LK0

�, as the link of the cone-point isP andP 2LK

0. Thus, in theabove diagram,

ŒP �ffl

�//

_

››

�.P / 6D 0:

0/

�

77

o

o

o

o

o

o

o

o

o

o

o

This argument shows that the desired diagonal arrow cannot exist for any col-lection LK

0 that contains any manifolds with nonzero signature. Thus wearenaturally led to consider only links with zero signature, that is, links whoseintersection form on middle dimensional homology possesses a Lagrangian sub-space. As you move along a stratum of odd codimension, these Lagrangiansubspaces should fit together, forming a subsheaf of the middle dimensionalcohomology sheafH associated to the link-bundle over the stratum. (Actually,no bundle neighborhood structure is required to do this.) Soa natural languagein which to phrase and solve the problem is sheaf theory.

From the sheaf-theoretic vantage point, the statement thata spaceX n doesnot satisfy the Witt condition means precisely that the canonical morphismIC �

Nm.X /! IC �

Nn.X / from lower to upper middle perversity is not an isomor-phism in the derived category. (We are using sheaves of real vector spaces nowand shall not indicate this further in our notation.) Thus there is no way to intro-duce a quadratic form whose signature one could take, using intersection chainsheaves. But one may ask how close to such sheaves one might get by usingself-dual sheaves onX . In [Ban02], we define a full subcategorySD.X / ofthe derived category onX , whose objectsS� satisfy all the axioms thatIC �

Nn.X /

satisfies, with the exception of the last axiom, the costalk vanishing axiom. Thisaxiom is replaced with the requirement thatS� be self-dual, that is, there is anisomorphismDS�Œn�Š S�, just as there is forIC �

Nm on a Witt space. Naturally,this category may be empty, depending on the geometry ofX . So we need todevelop a structure theorem forSD.X /, and this is done in [Ban02]. It turnsout that every such objectS� interpolates betweenIC �

Nm andIC �

Nn, i.e. possesses a

238 MARKUS BANAGL

factorizationIC �

Nm!S�! IC �

Nn of the canonical morphism. The two morphismsof the factorization are dual to each other. Note that in the basic two strata case,the mapping cone of the canonical morphism is the middle cohomology sheafH of the link-bundle. We prove that the mapping cone ofIC �

Nm! S�, restrictedto the stratum of odd codimension, is a Lagrangian subsheaf of H, so that thecircle to the above geometric ideas closes. The main result of [Ban02] is anequivalence of categories betweenSD.X / and a fibered product of categoriesof Lagrangian structures, one such category for each stratum of odd codimen-sion. This then is a kind of Postnikov system forSD.X /, encoding both theobstruction theory and the constructive technology to manufacture objects inSD.X /.

SupposeX is such thatSD.X / is not empty. An objectS� in SD.X / definesa signature�.S�/2Z by taking the signature of the quadratic form that the self-duality isomorphismDS�Œn�Š S� induces on the middle dimensional hyperco-homology group ofS�. Since restricting a self-dual sheaf to a transverse (to thestratification) subvariety again yields a self-dual sheaf on the subvariety, we get asignature for all transverse subvarieties and thus an L-classL�.S

�/2H�.X IQ/,using maps to spheres and Serre’s theorem as indicated in thebeginning. Weprove in [Ban06b] thatL�.S

�/, in particular�.S�/DL0.S�/, is independent of

the choice ofS� in SD.X /. Consequently, a non-Witt space has a well-definedL-classL�.X / and signature�.X /, providedSD.X / is not empty.

Let Sign.pt/ be the bordism group of pairs.X;S�/, whereX is a closedoriented topological or PLn-dimensional pseudomanifold andS� is an objectof SD.X /. Admissible bordisms are oriented compact pseudomanifolds-with-boundaryY nC1, whose interior intY is covered with an object ofSD.int Y /

which pushes to the given sheaf complexes on the boundary. These groups havebeen introduced in [Ban02] under the name˝SD

�. Let us compute these groups.

The signature.X;S�/‘ �.S�/ is a bordism invariant and hence induces a map� WSig4k.pt/!Z. This map is onto, since e.g..CP2k ;RCP2k Œ4k�/ (and disjointcopies of it) is in Sig4k.pt/. However, contrary for example to Witt bordism,� is also injective: Suppose�.X;S�/ D 0. Let Y 4kC1 be the closed cone onX . Define a self-dual sheaf on the interior of the punctured cone by pullingbackS� from X under the projection from the interior of the punctured cone,X�.0; 1/, toX . According to the Postnikov system of Lagrangian structures forSD.int Y /, the self-dual sheaf on the interior of the punctured cone will havea self-dual extension inSD.int Y / if, and only if, there exists a Lagrangianstructure at the cone-point (which has odd codimension4k C 1 in Y ). ThatLagrangian structure exists because�.X;S�/D 0. Let T� 2 SD.int Y / be anyself-dual extension given by a choice of Lagrangian structure. [email protected];T�/D

.X;S�/ and thusŒ.X;S�/�D 0 in Sig4k.pt/. Clearly, Sign.pt/D 0 for n 6� 0.4/


because an anti-symmetric form always has a Lagrangian subspace and the coneon an odd dimensional space is even dimensional, so in these cases there areno extension problems at the cone point — just perform a one-step Goresky–MacPherson–Deligne extension. In summary then, one has

Sign.pt/Š

�

Z; n� 0.4/,0; n 6� 0.4/.

(Note that in particular the de Rham invariant has been disabled and the signatureis a complete invariant for these bordism groups.) Minatta [Min04], [Min06]takes this as his starting point and constructs a bordism theory Sig�.�/, calledsignature homology, whose coefficients are the above groups Sig�.pt/. Elementsof Sign.Y / are represented by pairs.X;S�/ as above together with a continuousmap X ! Y . For a detailed proof that Sig�.�/ is a generalized homologytheory when PL pseudomanifolds are used, consult the appendix of [Ban06a].Signature homology is represented by an MSO module spectrumMSIG, whichis also a ring spectrum. Regarding a smooth manifold as a pseudomanifoldwith one stratum covered by the constant sheaf of rank1 concentrated in onedimension defines a natural transformation of homology theories˝SO

� .�/ !

Sig�.�/. Thus, MSIG is2-integrally a product of Eilenberg–Mac Lane spectra,

MSIG.2/ 'Y

j�0

K.Z.2/; 4j /:

As for the odd-primary situation, the isomorphism Sig�.pt/˝Z ZŒ12�! ZŒ1

2; t �

given byŒ.X 4k ;S�/�˝ a‘ a�.S�/tk , determines an identification

˝SO� .Y /˝˝SO

� .pt/ Sig�.pt/˝Z ZŒ12�

Š

�˝SO� .Y /˝˝SO

� .pt/ ZŒ12; t �:

A natural isomorphism of homology theories

˝SO�.Y /˝˝SO

� .pt/ Sig�.pt/˝Z ZŒ12�

Š

� Sig�.Y /˝Z ZŒ12�

is induced by sendingŒMf! Y �˝ Œ.X;S�/� to Œ.M �X;P�;M �X !M

f!

Y /�, whereP� is the pullback sheaf ofS� under the second-factor projection.Composing, we obtain a natural isomorphism

˝SO�.�/˝˝SO

� .pt/ ZŒ12; t �

Š

� Sig�.�/˝Z ZŒ12�;

describing signature homology at odd primes in terms of manifold bordism.

Again, it follows in particular that the natural map

˝SO� .Y /˝Z ZŒ1

2; t �! Sig�.Y /˝Z ZŒ1

2�

240 MARKUS BANAGL

is a surjection, which frequently allows one to reduce bordism invariant calcu-lations on non-Witt spaces to the manifold case. We observedthis in [Ban06a]and apply it there to establish a multiplicative characteristic class formula for thetwisted signature and L-class of non-Witt spaces. LetX n be a closed orientedWhitney stratified pseudomanifold and letS be a nondegenerate symmetric localsystem onX . If SD.X / is not empty, that is,X possesses Lagrangian structuresalong its strata of odd codimension so thatL�.X / 2H�.X IQ/ is defined, then

L�.X I S/D echŒS�K \L�.X /:

For the special case of the twisted signature�.X I S/DL0.X I S/, one has there-fore

�.X I S/D hechŒS�K ;L.X /i:

We shall apply the preceding ideas in defining aSullivan orientation�.X /2ko�.X /˝ZŒ1

2� for a pseudomanifoldX that possesses generalized Poincare du-

ality (that is, its self-dual perverse categorySD.X / is not empty), but need notsatisfy the Witt condition. In [Sul05], Sullivan defined foran oriented rationalPL homology manifoldM an orientation class�.M /2 ko�.M /˝ZŒ1

2�, whose

Pontrjagin character is the L-classL�.M /. For a Witt spaceX n, a Sullivan class�.X /2ko�.X /˝ZŒ1

2�was constructed by Siegel [Sie83], using the intersection

homology signature of a Witt space and transversality to produce the requisiteSullivan periodicity squares that represent elements ofKO4k.N; @N /˝ ZŒ1

2�,

whereN is a regular neighborhood of a codimension4k PL-embedding ofXin a high dimensional Euclidean space. An element inko4k.N; @N /˝ZŒ1

2� cor-

responds to a unique element inkon.X /˝ZŒ12� by Alexander duality. Siegel’s

isomorphism (3-1) is then given by the Hurewicz-type map

˝Witt�.Y /˝ZŒ1

2��ko�.Y /˝ZŒ1

2�

[Xf� Y ]˝ 1‘ f��.X /;

wheref� W ko�.X /˝ZŒ12�! ko�.Y /˝ZŒ1

2�. In particular, the transformation

(3-1) maps the Witt orientation classŒX �Witt 2 ˝Wittn .X /, given by the identity

mapf D idX WX ! X , to�.X /.

REMARK 5.1. In [CSW91], there is indicated an extension to continuous actionsof a finite groupG on a Witt spaceX . If the action satisfies a weak conditionon the fixed point sets, then there is a homeomorphism invariant class�G.X /

in the equivariant KO-homology ofX away from2, which is the Atiyah–SingerG-signature invariant for smooth actions on smooth manifolds.

Let P be a compact polyhedron. Using Balmer’s4-periodic Witt groups of tri-angulated categories with duality, Woolf [Woo08] defines groupsW c

� .P /, calledconstructible Witt groups ofP because the underlying triangulated categories


are the derived categories of sheaf complexes that are constructible with respectto the simplicial stratifications of admissible triangulations of P . (The dualityis given by Verdier duality.) Elements ofW c

n .P / are represented by symmet-ric self-dual isomorphismsd W S� ! .DS�/Œn�. The periodicity isomorphismW c

n .P /ŠW cnC4

.P / is induced by shifting such ad twice:

d Œ2� W S�Œ2�� .DS�

/Œn�Œ2�D .DS�/Œ�2�ŒnC 4�DD.S�

Œ2�/ŒnC4�:

(Shifting only once does not yield a correct symmetric isomorphism with respectto the duality fixed forW c

�.P /.) Woolf shows that for commutative regular

Noetherian ringsR of finite Krull dimension in which2 is invertible, for ex-ampleRD Q, the assignmentP ‘W c

�.P / is a generalized homology theory

on compact polyhedra and continuous maps. LetK be a simplicial complextriangulatingP . Relating both Ranicki’s.R;K/-modules on the one hand andconstructible sheaves on the other hand to combinatorial sheaves onK, Woolfobtains a natural transformation

L�.R/�.K/�W c

�.jKj/

(jKj DP ), which he shows to be an isomorphism when every finitely generatedR-module can be resolved by a finite complex of finitely generated freeR-modules. Again, this applies toR D Q. Given a mapf W X n ! P from acompact oriented Witt spaceX n into P , the pushforwardRf�.d/ of the sym-metric self-duality isomorphismd W IC �

Nm.X /ŠDIC �

Nm.X /Œn� defines an elementŒRf�.d/� 2W c

n .P /. This induces a natural map

˝Wittn .P /�W c

n .P /;

which is an isomorphism whenn> dimP . Given anyn� 0, we can iterate the4-periodicity untilnC 4� > dimP and obtain

W cn .P /ŠW c

nC4.P /Š � � � ŠW cnC4k.P /Š˝

WittnC4k.P /;

wherenC 4k > dimP . Thus, as Woolf points out, the Witt class of any sym-metric self-dual sheaf onP is given, after a suitable even number of shifts,by the pushforward of an intersection chain sheaf on some Witt space. Thisviewpoint also allows for the interpretation of L-classes as homology operationsW c

� .�/!H�.�/ or˝Witt� .�/!H�.�/. Other characteristic classes arising in

complex algebraic geometry can be interpreted through natural transformationsas well. MacPherson’s Chern class of a variety can be defined as the imagecM

� .1X / of the function1X under a natural transformationcM� WF.�/!H�.�/,

whereF.X / is the abelian group of constructible functions onX . The Baum–Fulton–MacPherson Todd class can be defined as the image tdBMF

� .OX / of OX

242 MARKUS BANAGL

under a natural transformation

tdBMF� WG0.�/!H�.�/˝Q;

whereG0.X / is the Grothendieck group of coherent sheaves onX . In [BSY],Brasselet, Schurmann and Yokura realized two important facts: First, there ex-ists a sourceK0.VAR=X / which possesses natural transformations to all threedomains of the characteristic class transformations mentioned. That is, thereexist natural transformations

K0.VAR=�/

xxq

q

q

q

q

q

q

q

q

q

››’’

N

N

N

N

N

N

N

N

N

N

N

F.�/ G0.�/ ˝Y.�/;

(5-1)

where Y.X / is the abelian group of Youssin’s bordism classes of self-dual con-structible sheaf complexes onX . That sourceK0.VAR=X / is the free abeliangroup generated by algebraic morphismsf W V ! X modulo the relation

ŒVf�X �D ŒV �Z

f j

�X �C ŒZf j

�X �

for every closed subvarietyZ � V . Second, there exists a unique natural trans-formation, themotivic characteristic class transformation,

Ty� WK0.VAR=X /�H�.X /˝QŒy�

such that

Ty�ŒidX �D Ty.TX /\ ŒX �

for nonsingularX , whereTy.TX / is Hirzebruch’s generalized Todd class ofthe tangent bundleTX of X . Characteristic classes for singular varieties areof course obtained by takingTy�ŒidX �. Under the above three transformations(5-1), ŒidX � is mapped to1X , ŒOX �, andŒQX Œ2 dimX �� (whenX is nonsingular),respectively. Following these three transformations withcM

�, tdBMF

�, and the

L-class transformation

˝Y.�/!H�.�/˝Q;

one obtainsTy� for y D �1; 0; 1, respectively. This, then, is an attractive uni-fication of Chern-, Todd- and L-classes of singular complex algebraic varieties,see also Yokura’s paper in this volume, as well as [SY07].

The natural transformation

˝Witt� .�/˝ZŒ1

2�� Sig�.�/˝ZŒ1

2�;


given by covering a Witt spaceX with the middle perversity intersection chainsheafS� D IC �

Nm.X /, which is an object ofSD.X /, is an isomorphism becauseon a point, it is given by the signature

˝Witt4k .pt/˝ZŒ1

2�ŠL4k.Q/˝ZŒ1

2�Š ZŒ1

2�Š Sig4k.pt/˝ZŒ1

2�

(the infinitely generated torsion ofL4k.Q/ is killed by inverting2), and

˝Wittj .pt/D 0D Sigj .pt/

for j not divisible by4. Inverting this isomorphism and composing with Siegel’sisomorphism (3-1), we obtain a natural isomorphism of homology theories

D W Sig�.�/˝ZŒ12�

Š

� ko�.�/˝ZŒ12�:

Let X n be a closed pseudomanifold, not necessarily a Witt space, but still sup-porting self-duality, i.e.SD.X / is not empty. Choose a sheafS� 2 SD.X /.Then the pair.X;S�/, together with the identity mapX!X , defines an elementŒX �Sig2 Sign.X /.

DEFINITION 5.2. Thesignature homology orientation classof ann-dimensionalclosed pseudomanifoldX with SD.X / 6D ?, but not necessarily a Witt space,is the elementŒX �Sig2 Sign.X /.

PROPOSITION5.3. The orientation classŒX �Sig is well-defined, that is, inde-pendent of the choice of sheafS� 2 SD.X /.

PROOF. Let T� 2 SD.X / be another choice. In [Ban06b], a bordism.Y;U�/,U� 2 SD.int Y /, is constructed between.X;S�/ and .X;T�/. Topologically,Y is a cylinderY Š X � I , but equipped with a nonstandard stratification, ofcourse. The identity mapX !X thus extends over this bordism by taking

Y !X; .x; t/‘ x: ˜

DEFINITION 5.4. TheSullivan orientationof ann-dimensional closed pseudo-manifoldX with SD.X / 6D?, but not necessarily a Witt space, is defined as

�.X /DD.ŒX �Sig˝ 1/ 2 ko�.X /˝ZŒ12�:

Let us compare signature homology andL�-homology away from2, at 2, and

rationally, following [Epp07] and drawing on work of Taylorand Williams,[TW79]. For a spectrumS , let S.odd/ denote its localization at odd primes. Wehave observed above that

MSIG.2/ 'Y

j�0

K.Z.2/; 4j /

244 MARKUS BANAGL

and, according to [Epp07] and [Min04],

MSIG.odd/ ' bo.odd/:

Rationally, we have the decomposition

MSIG˝Q'Y

j�0

K.Q; 4j /:

Thus MSIG fits into a localization pullback square

MSIGloc.odd/

//

loc.2/

››

bo.odd/

››

Q

K.Z.2/; 4j /�

//

Q

K.Q; 4j /:

The symmetric L-spectrumL� is an MSO module spectrum, so it is2-integrallya product of Eilenberg–Mac Lane spectra,

L�

.2/ 'Y

j�0

K.Z.2/; 4j /�K.Z=2; 4j C 1/:

Comparing this to MSIG.2/, we thus see the de Rham invariants coming in.Away from 2, L

� coincides withbo,

L�

.odd/ ' bo.odd/;

as does MSIG. Rationally,L� is again

L�˝Q'

Y

j�0

K.Q; 4j /:

ThusL� fits into a localization pullback square

L�

loc.odd///

loc.2/

››

bo.odd/

››

Q

K.Z.2/; 4j /�K.Z=2; 4j C 1/�0

//

Q

K.Q; 4j /:

The map� factors as

Y

j�0

K.Z.2/; 4j /�Œ

Y

j�0

K.Z.2/; 4j /�K.Z=2; 4j C 1/�0

�Y

j�0

K.Q; 4j /;


where� is the obvious inclusion, not touching the2-torsion nontrivially. Hence,by the universal property of a pullback, we get a map� from signature homologyto L

�-homology,

MSIG

loc.2/

››

�

**

loc.odd/

++

L�

loc.odd/

//

loc.2/

››

bo.odd/

››

Q

K.Z.2/; 4j / ffl� �

//

�

44

Q

K.Z.2/; 4j /�K.Z=2; 4j C 1/�0

//

Q

K.Q; 4j /:

On the other hand,�0 factors as

Y

j�0

K.Z.2/; 4j /�K.Z=2; 4j C 1/proj�

Y

j�0

K.Z.2/; 4j /��

Y

j�0

K.Q; 4j /;

where proj is the obvious projection. Again using the universal property of apullback, we obtain a map� W L�!MSIG. The map� is a homotopy splittingfor �, ��' id, since projı�D id. It follows that via�, signature homology isa direct summand in symmetricL�-homology. We should like to point out thatthe diagram

MSO //

››

MSIG�

››

MSTOP //

L�

doesnot commute. This is essentially due to the fact that the de Rham invariantis lost in Sig�, but is still captured inL�

�. In more detail, consider the induced

diagram on�5,

˝SO5.pt/ //

››

Sig5.pt/D 0

�››

˝STOP5

.pt/ // L�

5.pt/:

The clockwise composition in the diagram is zero, but the counterclockwisecomposition is not. Indeed, letM 5 be the Dold manifoldP .1; 2/ D .S1 �

CP2/=.x; z/� .�x; Nz/. Its cohomology ring withZ=2-coefficients is the sameas the one of the untwisted product, that is, the truncated polynomial ring

Z=2Œc; d �=.c2 D 0; d3 D 0/;

246 MARKUS BANAGL

wherec has degree one andd has degree two. The total Stiefel–Whitney classof M 5 is

w.M /D .1C c/.1C cC d/3;

so that the de Rham invariant dR.M / is given by dR.M /Dw2w3.M /D cd2,which is the generator. We also see thatw1.M /D0, so thatM is orientable. Thecounterclockwise composition maps the bordism class of a smooth5-manifoldto its de Rham invariant inL�

5.pt/ D L5.Ze/ D Z=2. The Dold manifoldM 5

represents the generatorŒM 5� 2 ˝SO5.pt/ D Z=2. Thus the counterclockwise

composition is the identity mapZ=2!Z=2 and the diagram does not commute.MappingM 5 to a point and using the naturality of the assembly map induces acommutative diagram

Sig5.M /�.M /

//

››

L�

5.M /

A//

››

L5.Z�1M /

"

››

0D Sig5.pt/�.pt/

// L�

5.pt/ Š

A// L5.Ze/D Z=2;

which shows that the signature homology orientation class of M , ŒM �Sig 2

Sig5.M / does not hit theL�-orientation ofM , ŒM �L 2 L�

5.M / under�, for

otherwise

0D "A�ŒM �SigD "AŒM �L D "��.M /D dR.M / 6D 0:

Thus one may take the viewpoint that it is perhaps not prudentto call�ŒX �Sig

an “L�-orientation” of a pseudomanifoldX with SD.X / not empty. Nor mighteven its image under assembly deserve the title “symmetric signature” ofX . Onthe other hand, one may wish to attach higher priority to the bordism invariance(in the singular world) of a concept such as the symmetric signature than to itscompatibility with manifold invariants and nonsingular bordism invariance, andtherefore deem such terminology justified.

We conclude with a brief remark on integral Novikov problems. Let � bea discrete group and letK.�; 1/ be the associated Eilenberg–Mac Lane space.The composition of the split inclusion Sign.K.�; 1//Œ L

�

n.K.�; 1// with theassembly map

A W L�

n.K.�; 1//!Ln.Z�/

yields what one may call a “signature homology assembly” map

ASig W Sign.K.�; 1//!Ln.Z�/;


which may be helpful in studying an integral refinement of theNovikov conjec-ture, as suggested by Matthias Kreck: When is the integral orientation class

˛�ŒM �Sig2 Sign.K.�; 1//

homotopy invariant? HereM n is a closed smooth oriented manifold with fun-damental group� D �1.M /; the map WM !K.�; 1/ classifies the universalcover ofM . Note that when tensored with the rationals, one obtains theclassicalNovikov conjecture because rationally the signature homology orientation classŒM �Sig is the L-classL�.M /. One usually refers to integral refinements such asthis one as “Novikovproblems” because there are groups� for which they areknown to be false.

References

[BC84] A. Borel et al.,Intersection cohomology, Progr. Math.50, Birkhauser, Boston,1984.

[Ban02] M. Banagl,Extending intersection homology type invariants to non-Witt spaces,Memoirs Amer. Math. Soc.160(2002), 1–83.

[Ban06a] , Computing twisted signatures and L-classes of non-Witt spaces,Proc. London Math. Soc. (3)92 (2006), 428–470.

[Ban06b] , The L-class of non-Witt spaces, Annals of Math.163:3 (2006), 743–766.

[Ban06c] , On topological invariants of stratified maps with non-Witt target,Trans. Amer. Math. Soc.358:5 (2006), 1921–1935.

[Ban07] , Topological invariants of stratified spaces, Springer Monographs inMathematics, Springer, Berlin, 2007.

[Ban09] , Intersection spaces, spatial homology truncation, and string theory,Lecture Notes in Math.1997Springer, Berlin, 2010.

[BB08] B. Bleile and H. J. Baues,Poincare duality complexes in dimension four,preprint arXiv:0802.3652v2, 2008.

[BCS03] M. Banagl, S. E. Cappell, and J. L. Shaneson,Computing twisted signaturesand L-classes of stratified spaces, Math. Ann.326:3 (2003), 589–623.

[Bre97] G. E. Bredon,Sheaf theory, second ed., Grad. Texts in Math.170, Springer,New York, 1997.

[BSY] J.-P. Brasselet, J. Schurmann, and S. Yokura,Hirzebruch classes and motivicChern classes for singular spaces, preprint math.AG/0503492.

[Che79] J. Cheeger,On the spectral geometry of spaces with cone-like singularities,Proc. Natl. Acad. Sci. USA76 (1979), 2103–2106.

[Che80] , On the Hodge theory of Riemannian pseudomanifolds, Proc. Sympos.Pure Math.36 (1980), 91–146.

248 MARKUS BANAGL

[Che83] , Spectral geometry of singular Riemannian spaces, J. DifferentialGeom.18 (1983), 575–657.

[CS91] S. E. Cappell and J. L. Shaneson,Stratifiable maps and topological invariants,J. Amer. Math. Soc.4 (1991), 521–551.

[CSW91] S. E. Cappell, J. L. Shaneson, and S. Weinberger,Classes topologiquescaracteristiques pour les actions de groupes sur les espaces singuliers, C. R. Acad.Sci. Paris Ser. I Math.313(1991), 293–295.

[Cur92] S. Curran,Intersection homology and free group actions on Witt spaces,Michigan Math. J.39 (1992), 111–127.

[CW91] S. E. Cappell and S. Weinberger,Classification de certaines espaces stratifies,C. R. Acad. Sci. Paris Ser. I Math.313(1991), 399–401.

[Epp07] T. Eppelmann,Signature homology and symmetric L-theory, Ph.D. thesis,Ruprecht-Karls Universitat Heidelberg, 2007.

[GM80] M. Goresky and R. D. MacPherson,Intersection homology theory, Topology19 (1980), 135–162.

[GM83] , Intersection homology II, Invent. Math.71 (1983), 77–129.

[Gor78] M. Goresky,Triangulation of stratified objects, Proc. Amer. Math. Soc.72(1978), 193–200.

[Gor84] , Intersection homology operations, Comm. Math. Helv.59 (1984),485–505.

[GS83] R. M. Goresky and P. H. Siegel,Linking pairings on singular spaces, Comm.Math. Helv.58 (1983), 96–110.

[Hen77] H. Hendriks,Obstruction theory in 3-dimensional topology: An extensiontheorem, J. London Math. Soc. (2)16 (1977), 160–164.

[KW06] F. Kirwan and J. Woolf,An introduction to intersection homology theory,second ed., Chapman & Hall/CRC, 2006.

[Min04] A. Minatta, Hirzebruch homology, Ph.D. thesis, Ruprecht-Karls UniversitatHeidelberg, 2004.

[Min06] , Signature homology, J. Reine Angew. Math.592(2006), 79–122.

[Par90] W. L. Pardon,Intersection homology, Poincare spaces and the characteristicvariety theorem, Comment. Math. Helv.65 (1990), 198–233.

[Ran79] A. A. Ranicki,The total surgery obstruction, Proceedings 1978Arhus Topol-ogy Conference, Lecture Notes in Math.763, Springer, 1979, pp. 275–316.

[Ran92] , Algebraic L-theory and topological manifolds, Cambridge Tracts inMath.102, Cambridge University Press, 1992.

[Sie83] P. H. Siegel,Witt spaces: A geometric cycle theory for KO-homology at oddprimes, Amer. J. Math.105(1983), 1067–1105.

[Sul70] D. Sullivan,A general problem, p. 230, Manifolds, Amsterdam, Lecture Notesin Math.197, Springer, Berlin, 1970.


[Sul05] Dennis P. Sullivan,Geometric topology: localization, periodicity and Galoissymmetry, K-Monographs in Mathematics8, Springer, Dordrecht, 2005 (reprint of1970 MIT notes. with a preface by Andrew Ranicki).

[SY07] J. Schurmann and S. Yokura,A survey of characteristic classes of singularspaces, Singularity Theory, World Scientific, 2007, pp. 865–952.

[Tho58] R. Thom,Les classes caracteristiques de Pontrjagin des varietes triangulees,Symposium Internacional de Topologia Algebraica, La Universidad Nacional Au-tonoma de Mexico y la Unesco, 1958, pp. 54–67.

[Tur90] V. G. Turaev,Three-dimensional Poincare complexes: Homotopy classificationand splitting, Russ. Acad. Sci., Sb.67 (1990), 261–282.

[TW79] L. Taylor and B. Williams,Surgery spaces: Formulae and structure, Proceed-ings 1978 Waterloo Algebraic Topology Conference, LectureNotes in Math.741,Springer, 1979, pp. 170–195.

[Wei94] S. Weinberger,The topological classification of stratified spaces, ChicagoLectures in Math., Univ. of Chicago Press, Chicago, 1994.

[Whi40] J. H. C. Whitehead,OnC 1-complexes, Ann. of Math.41 (1940), 809–824.

[WJ66] R. E. Williamson Jr.,Cobordism of combinatorial manifolds, Ann. of Math.83(1966), 1–33.

[Woo08] J. Woolf,Witt groups of sheaves on topological spaces, Comment. Math.Helv. 83:2 (2008), 289–326.

MARKUS BANAGL

MATHEMATISCHES INSTITUT

UNIVERSITAT HEIDELBERG

IM NEUENHEIMER FELD 28869120 HEIDELBERG

GERMANY

[email protected]


Intersection homology Wang sequenceFILIPP LEVIKOV

ABSTRACT. We prove the existence of a Wang-like sequence for intersectionhomology. A result is given on vanishing of the middle dimensional inter-section homology group of “generalized Thom spaces”, whichnaturally occurin the decomposition formula of S. Cappell and J. Shaneson. Based upon thisresult, consequences for the signature are drawn.

For non-Witt spacesX , signature and L-classes are defined via the hyper-cohomology groupsHi.X I IC �

L/, introduced in [Ban02]. A hypercohomology

Wang sequence is deduced, connectingHi.�I IC �

L/ of the total space with

that of the fibre. Also here, a consequence for the signature under collapsingsphere-singularities is drawn.

1. Introduction

The goal of this article is to add to the intersection homology toolkit anotheruseful long exact sequence. In [Wan49], H. C. Wang, calculating the homol-ogy of the total space of a fibre bundle over a sphere, actuallyproved an exactsequence, which is named after him today. It is a useful tool for dealing withfibre bundles over spheres and it is natural to ask: Is there a Wang sequence forintersection homology?

Given an appropriate notion of a stratified fibration, the natural framework fordealing with a question of the kind above would be an intersection homologyanalogue of a Leray–Serre spectral sequence. Greg Friedmanhas investigatedthis and established an appropriate framework in [Fri07]. For a simplified settingof a stratified bundle, however, i.e., a locally trivial bundle over a manifold witha stratified fibre, it seems more natural to explore the hypercohomology spectralsequence directly. In the following we are going to demonstrate this approach.

Mathematics Subject Classification:55N33.Keywords:intersection homology, Wang sequence, signature.

251

252 FILIPP LEVIKOV

Section 3 is a kind of foretaste of what is to come. We prove themonodromycase by hand using only elementary intersection homology and apply it to calcu-late the intersection homology groups of neighbourhoods ofcircle singularitieswith toric links in a 4-dimensional pseudomanifold.

We recall the construction of induced maps in Section 4.1. Because of theircentral role in the application, the Cappell–Shaneson decomposition formula isexplained in Section 4.2. Section 5 contains a proof of the Wang sequence forfibre bundles over simply connected spheres. It is shown thatunder a certainassumption the middle-dimensional middle perversity intersection homologyof generalized Thom spaces of bundles over spheres vanish. The formula ofCappell and Shaneson then implies, that in this situation the signature does notchange under the collapsing of the spherical singularities.

In Section 6, we demonstrate a second, concise proof — this ismerely thesheaf-theoretic combination of the relative long exact sequence and the suspen-sion isomorphism. However, this proof is mimicked in Section 7 to derive aWang-like sequence for hypercohomology groupsH�.X IS�/ with values in aself-dual perverse sheaf complexS� 2 SD.X /. In Section 8, finally, togetherwith Novikov additivity, this enables us to identify situations when collapsingspherical singularities in non-Witt spaces does not changethe signature.

2. Basic notions

We will work in the framework of [GM83]. In the followingX D Xn �

Xn�2 � � � � � X0 � X�1 D ? will denote an orientedn-dimensional strati-fied topological pseudomanifold. The intersection homology groups ofX withrespect to perversityNp are denoted byIH

Npi .X /, and the analogous compact-

support homology groups byIHc Npi .X /. The indexing convention is also that of

[GM83]. Most of the fibre bundles to be considered in the following are goingto bestratifiedbundles in the following sense (see also [Fri07, Definition 5.6]):

DEFINITION 2.1. A projectionE!B to a manifold is called a stratified bundleif for each pointb 2 B there exist a neighbourhoodU � B and a stratum-preserving trivializationp�1.U / Š U �F , whereF is a topological stratifiedpseudomanifold.

We will also restrict the automorphism group ofF to stratum preserving auto-morphisms and work with the corresponding fibre bundles in the usual sense.Since we will basically need the local triviality, Definition 2.1 is mostly suf-ficient. When we pass to applications for Whitney stratified pseudomanifolds,however, the considered bundles will actually be fibre bundles — this followsfrom the theory of Whitney stratifications. A stratificationof the fibre inducesan obvious stratification of the total space with the samel-codimensional links,

INTERSECTION HOMOLOGY WANG SEQUENCE 253

namely byEkCn�l — the total spaces of bundles with fibreFk�l and n thedimension ofB.

3. Mapping torus

PROPOSITION3.1 (INTERSECTION HOMOLOGYWANG SEQUENCE FORS1).Let F D Fn � Fn�2 � � � � � F0 be a topological stratified pseudomanifold,� WF!F a stratum and codimension preserving automorphism, i.e., a stratumpreserving homeomorphism with stratum preserving inversesuch that both mapsrespect the codimension. Let M� be the mapping torus of�, i.e., the quotientspaceF � I=.y; 1/s .�.y/; 0/. Denote byi W F D F � 0ŒM� the inclusion.Then the sequence

� � �� IHc Np

k.F /

id ��

��! IHc Np

k.F /

i�� IH

c Np

k.M�/

@� IH

c Np

k�1.F /��

is exact.

PROOF. The proof is analogous to the one for ordinary homology. Start with thequotient mapq W .F � I;F /! .M� ;F / and look at the corresponding diagramof long exact sequences of pairs. The boundary ofF � I is a codimension 1stratum and hence not a pseudomanifold. We have either to introduce the notionof a pseudomanifold with boundary here or work with intersection homologyfor cs-sets [Kin85; HS91]. However, we can also manage with awork-around:Define

I" WD .�"; 1C "/; @I" WD .�"; "/[ .1� "; 1C "/; F" WD F � .�"; "/:

We extend the identification.y; 1/ s .�.y/; 0/ to F � I" by introducing thequotient mapq W F � I"!M� ,

q.y; t/D

8

<

:

.��1.y/; 1C t/ if t2 .�"; 0�

.y; t/ if t2 .0; 1/

.�.y/; t � 1/ if t2 Œ1; 1C "/:

Evidently,M� D q.F �I"/. Now F �@I"D .F �@I"/nC1 � .F �@I"/� � � � �

.F�@I"/0 is an opensubpseudomanifold ofF�I" andF DFn�Fn�2� � � � �

F0 sits normally nonsingular inM� . Hence the inclusions induce morphismson intersection homology and we get a morphism of the corresponding exactsequences of pairs:

: : : IHc Np

k.F�I";F�@I"/ IH

c Np

k�1.F�@I"/ IH

c Np

k�1.F�I"/ : : :

: : : IHc Np

k.M� ;F"/ IH

c Np

k�1.F"/ IH

c Np

k�1.M�/ : : :............................................................................................................

............

i�

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

q�

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

q�

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

q�

....................................................................

............

0

..................................................................................................

............

...............................................................

............

0

......................................................................

............

..................................................................................................

............

@

....................................................................................................................................................

............

@

................................................................................

............

j�

254 FILIPP LEVIKOV

The “boundary” ofF�I" is the disjoint union of two components of the formF �R, soj� is surjective and the outer arrows are zero maps. The connectingmorphism@ is injective and therefore an isomorphism onto its image, i.e., onto

kerj�

D˚

.˛; ˇ/ j˛ 2 IHc Np

k.F�.�";C"//; ˇ 2 IH

c Np

k.F�.1�"; 1C"//; Œ˛Cˇ�D 0

Df.˛;�˛/gŠ IHc Np

k.F�R/Š IH

c Np

k.F /:

The middleq� maps.˛;�˛/ to .˛��.˛// 2 IHc Np

k.F"/Š IH

c Np

k.F /. Sinceq

commutes with@, one has

@ ı q� ı @j�1 D q�jkerj�ŠIH

c Np

k.F / D id��:

Hence, we have the diagram

: : : IHc Np

k.F / IH

c Np

k.F / IH

c Np

k.M�/ IH

c Np

k.M� ;F"/

ko

kerj�

IHc Np

kC1.F � I";F � @I"/ IH

c Np

kC1.M� ;F"/

........................................................................................................................................................................

............

q�jkerj�.............................................

............

i�......................................

............

...........................................

............

q�

............................

............

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.

....

.

.

.

.

.

.

.

.

....

@..................................................

.

.

.

.

.

.

..

....@j�1..................................................

.

.

.

.

.

.

..

....o

where the top sequence is exact and the bottom square is commutative. As in ord-inary homology one can show thatq� WIH

c Np

k.F�I";F�@I"/!IH

c Np

k.M� ;F"/

is an isomorphism. Finally, observe that on the right hand sideIHc Np

k.M� ;F"/Š

IHc Np

k�1.F / via @ ı q�1

�. ˜

Let X DX4 �X1 �X0 be a compact stratified pseudomanifold withX0 D?.Then, the stratum of codimension 3 is just a disjoint union ofcircles X1 D

S1 t � � � t S1. If we assumeX to be PL, the linkL at a pointp 2 X1 isindependent ofp within a connective component ofX1. Furthermore, inX4,there is a neighbourhoodU of the circle containingp, which is a fibre bundleoverS1 and hence homeomorphic to the mapping torusM� with

� W c.L/Š

! c.L/:

Putting this data together and using the Wang sequence we cancomputeIH

c Np

k.U /, with U a neighbourhood ofX1 � X4. The groupIH

c Np

k.X / can

then be computed via the Mayer–Vietoris sequence.In this section we restrict ourselves to the case ofL being a torusT 2. While

the orientation preserving mapping class group of the torusis known to beSL.2IZ/, we have to make the following restriction on its cone: In thefollowing,we look only at those automorphisms� W c.T 2/! c.T 2/ which are induced


by an automorphism of the underlying torus W T 2 Š

�! T 2.1 It is given by amatrix˛ 2SL.2IZ/ and by abuse of notation we will again write˛ for this torusautomorphism.

Defining�k to be the mapIHc Np

k.c.T 2//

id �.c.˛//��!IH

c Np

k.c.T 2//, we obtain

the sequence

� � �� IHc Np

k.c.T 2//

�k�

IHc Np

k.c.T 2//

i�� IH

c Np

k.M˛/

@� IH

c Np

k�1.c.T 2//��

For the open cone we have

IHc Np

k.c.T 2//D

8

<

:

IHc Np0.T 2/ for k D 0;

IHc Np1.T 2/ for Np D N0 andk D 1;

0 else:

Clearly, IHc Np

k.M˛/ D 0 for k � 3. Now examine the nontrivial part of the

sequence

0� IHc Np2.M˛/

@� IH

c Np1.c.T 2//

�1� IH

c Np1.c.T 2//

i��

IHc Np1.M˛/

@� IH

c Np0.c.T 2//

�0� IH

c Np0.c.T 2//

i�� IH

c Np0.M˛/� 0:

Sincec.˛/0 maps a point to a point, clearlyc.˛/0D id, hence�0D id� idD 0.It follows thatIH

c Np0.M˛/D Z.

Note that the only possible perversities in this example areN0 and Nt . So farwe have not distinguished between them. Due toIH c Nt

1 .c.T2// D 0 there is

IH c Nt2 .M˛/D 0 andIH c Nt

1 .M˛/D Z. For the zero perversity, we have

0� IH c N02 .M˛/

@��! IH c N0

1 .c.T2//

�1�! IH c N0

1 .c.T2//� IH c N0

1 .M˛/�

ko

Z ˚ZIH c N0

0 .c.T2//� 0

The groupIH c N01 .c.T

2// is isomorphic toH1.T2/ and is therefore generated

by the corresponding homology classes of the torus. Hence,.c.˛//� is justthe matrix˛. If ˛ D id; �1 D 0 and IH c N0

2 .M˛/ Š Z ˚ Z. In the generalcase,IH c N0

2 .M˛/ is isomorphic to im@� D ker�1. We examine the determinant

1I believe that in the PL context this does not constitute a real restriction. With [Hud69, Theorem 3.6C]we can find an admissible triangulation ofc.T 2/, such that� becomes simplicial. Furthermore, it is notdifficult to see, that the simplicial link of the cone pointL.c/ is preserved under�. Since the geometricrealization ofL.c/ is a torus, we get a candidate for . By linearity, every slice betweenL.c/ and the conepoint c is mapped by . If we could extend the argument to the rest ofc.T 2/ the goal would be achieved.

256 FILIPP LEVIKOV

of the matrix: det�1 D det.id�˛/ D p˛.1/, wherep˛.t/ is the characteristicpolynomial of˛, which is

p˛.t/D det�

t id��

a11a21

a12a22

��

D .t � a11/.t � a22/� a21a12

D t2� .a11C a22/t C .a11a22� a21a12/

D t2� tr˛ t C det˛

D t2� tr˛ t C 1:

Here, tr is the trace of 2 SL.2IZ/. Thus we have det�1 D 2� tr˛ and get

IH c N02 .M˛/Š

8

<

:

Z˚Z if ˛ D id;Z if tr ˛ D 2 and˛ ¤ id;0 if tr ˛ ¤ 2:

SinceIH c N00 .c.T

2// is free, the sequence above reduces to a split short exactsequence

0� coker�1� IH c N01 .M�/� IH c N0

0 .c.T2//� 0:

HenceIH c N01 .M�/ Š Z˚ coker�1. In this final case our interest reduces to a

cokernel calculation of the2� 2-matrix�1 D id�˛. The image im�1 � Z˚Z

is of the formnZ˚mZ; n;m 2 Z and so every groupZ˚Z=nZ˚Z=mZ canbe realized asIH c N0

1 .M�/. In particular a torsion intersection homology groupmay appear. Using det�1D det.id�˛/D 2� tr ˛ as above, we immediately seethat

coker�1 Š

�

Z˚Z if ˛ D id;0 if tr ˛ D 1; 3:

Summarizing all these results we get:

PROPOSITION3.2. Let M˛ be the mapping torus over the open conec.T 2/ of

a torus glued via W T 2 Š

�! T 2. Then its intersection homology groups are

IH c Ntk .M˛/Š

�

Z if k D 0; 1;

0 if k � 2;

IH c N0k .M˛/Š

8

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

:

Z if k D 0;

Z˚Z˚Z if k D 1 and˛ D idZ if k D 1 and tr˛ D 1; 3;

Z˚ coker.id�˛/ if k D 1 (in general);Z if k D 2; tr˛ D 2 and˛ ¤ id;Z˚Z if k D 2 and˛ D id;0 if k D 2 and tr˛ ¤ 2;

0 if k � 3:


EXAMPLE 3.3. LetX be the fibre bundle overS1 with fibre˙T 2 and mon-odromy˛ 2 SL2.Z/. We assume ˛ to be orientation preserving, i.e., the sus-pension points are fixed under it. The spaceX has a filtrationX4�X1DS1tS1

and our situation applies. Let be�

21

�10

�

. First, using the ordinary Wangsequence, we compute the homology of the total spaceE of the fibre bundleT 2!E! S1 with the same monodromy:

0!H3.E/@�!H2.T

2/id �˛��!H2.T

2/!H2.E/@�!H1.T

2/

id �˛��!H1.T

2/!H1.E/@�!H0.T

2/id �˛��!H0.T

2/!H0.E/! 0:

In degrees 2 and 0, the map˛� is the identity, so we substitute zeros for id�˛�

to see thatH3.E/Š ZŠH0.E/. In degree 1, the map� is just the matrix .Using im@2 D ker.id�˛�/Š Z, we get the sequence

0!H2.T2/!H2.E/

@�! Z! 0;

which yieldsH2.E/Š Z˚Z and

0! coker.id�˛�/!H1.E/! Z! 0:

It follows by coker.id�˛�/ Š Z that H1.E/ Š Z˚ Z. Let us now computethe intersection homology groups ofX via the Mayer–Vietoris sequence. Theneighbourhoods of the twoS1 are of the desired form, i.e., mapping tori overc.T 2/ and their intersection is a fibre bundle overS1 with fibre T 2�R, so thatthe intersection homology groups are justH�.E/ from above. Looking at theexact sequence

0� IHc Np4.X /! IH

c Np3.E/!IH

c Np3.M˛/˚IH

c Np3.M˛/! � � � ;

jj jj

0 0

we see thatIHc Np4.X /ŠH3.E/ŠZ. In degree 0 the inclusion ofE induces an

injection on homology, i.e.,

0�H0.E/! IHc Np0.M˛/˚ IH

c Np0.M˛/! IH

c Np0.X /! 0;

andIHc Np0.X /ŠZ as it should be. Turning to the interesting degrees, we look at

Np D .0; 1; : : :/ first. Due toIH c Nt2 .M˛/D 0, there isIH c Nt

2 .X /Š ker.i�˚ i�/1.Finally IH c Nt

1 .X / is isomorphic to the cokernel of the inclusion.i� ˚ i�/1 W

H1.E/! IH c Nt1 .M˛/˚IH c Nt

1 .M˛/ŠZ˚Z, which is the diagonal map; henceIH c Nt

1 .X /ŠZ andIH c Nt2 .X /Š ker.i�˚ i�/1ŠZ. Similarly, for NpD .0; 0; : : :/

we haveIH c N03 .X / Š ker.i�˚ i�/2 Š Z; with IH c N0

1 .M˛/ Š Z˚Z it followsIH c N0

1 .X / Š Z˚ Z. And IH c N02 .X / Š coker.i� ˚ i�/2 Š Z. Because all the

258 FILIPP LEVIKOV

groups are free, the duality is already seen working with integral coefficients,especially

IH c N03 .X /Š IH c Nt

1 .X /Š Z˚Z;

IH c N01 .X /Š IH c Nt

3 .X /Š Z:

4. Some more advanced tools

4A. Normally nonsingular maps. Intersection homology is not a functor on thefull subcategory ofTop consisting of pseudomanifolds, since induced maps donot exist in general. However, on the category of topological pseudomanifoldsand normally nonsingular maps, intersection homology is a bivariant theory inthe sense of [FM81]. This fact is often suppressed. Since most of the mapswhich we will encounter are normally nonsingular, we recallin this section howinduced maps are constructed. See particularly [GM83, 5.4].

DEFINITION 4.1. A mapf W Y ! X between two pseudomanifolds is callednormally nonsingular (nns) of relative dimensionc D c1 � c2 if it is a com-position of a nns inclusion of dimensionc1 — meanining thatY is sitting in ac1-dimensional tubular neighbourhood in the target — and a nnsprojection, i.e.,a bundle projection withc2-dimensional manifold fibre.

EXAMPLE 4.2. An open inclusionU ŒX is normally nonsingular. The inclu-sion of the fibreF D b�FŒE, whereE is fibred over a manifold is normallynonsingular. The projectionRn �X ! X is normally nonsingular.

PROPOSITION4.3 [GM83, 5.4.1, 5.4.2].Let f W Y ! X be normally non-singular of codimensionc. Then there are isomorphisms

f � IC �

Np.X /Š IC �

Np.Y /Œc� and f ! IC �

Np.X /Š IC �

Np.Y /:

DEFINITION 4.4. If f W Y ! X is a proper normally nonsingular map ofcodimensionc, we have induced homomorphisms

f� W IHNp

k.Y /! IH

Np

k.X / and f � W IH

Np

k.X /! IH

Np

k�c.Y /:

They are constructed by considering the adjunction morphisms of the adjointpairs.Rf!; f

!/ and.f �;Rf�/,

Rf!f! IC �

Np.X /! IC �

Np.X / and IC �

Np.X /!Rf�f� IC �

Np.X /;

by combining them with the proposition above and by finally applying hyper-cohomology.

We will also need induced maps on intersection homology withcompact sup-ports, which are not discussed in [GM83]. The above construction of Goreskyand MacPherson works equally well forIH

c Np� . If f is not proper, the map


f � still exists. For the case of compact supports,f! W IHc Np

k.Y /! IH

c Np

k.X /

can be constructed in the same manner. These different maps are listed in thefollowing table — note that onlyf� andf � for properf are explicitly men-tioned in [GM83, 5.4].

f proper f not proper

f� W IHNp

k.Y /! IH

Np

k.X /

f! W IHc Np

k.Y /! IH

c Np

k.X / f! W IH

c Np

k.Y /! IH

c Np

k.X /

f � W IHNp

k.X /! IH

Np

k�c.Y / f � W IH

Np

k.X /! IH

Np

k�c.Y /

f ! W IHc Np

k.X /! IH

c Np

k�c.Y /

4B. Behaviour under stratified maps. Computing intersection homology in-variants of one space out of the invariants of the other oftenrelies on the de-composition formula of S. Cappell and J. Shaneson [CS91]. Since we will needit in the application below, we briefly recall it in this section.

Let f W X n ! Y m be a stratified map between closed, oriented Whitneystratified sets of even relative dimension2t D n � m, Y having only even-codimensional strata. LetS� 2Db

c .X / be a self-dual complex. Denote byV theset of components of pure strata ofY . For eachy 2 Vy 2 V , define2

Ey WD f�1.c L.y//[f �1L.y/ cf �1.L.y//;

whereL.y/ is the link of the stratum componentVy containingy. If y lies inthe top stratum, we setEy D f

�1.y/. We have the inclusions

Ey

iy‹ f �1.N .y//

�y

ŒX;

whereN.y/ is the normal slice ofy. Note thatN.y/Š c L [email protected]/ŠL.y/

(see [GM88] or [Ban07, 6.2]). Define now the complex

S�.y/D �cone��c�t�1Riy��

!yS�;

where�cone�

stands for truncation over the cone point3 of cf �1.L.y// and2cD

2c.V /D n�dimV is the codimension ofV .

2Here,c L stands for the closed coneL � Œ0; 1� =L � f0g.3There is a general notion of truncation over a closed subset in [GM83, 1.14]. Letc be the cone-point.

For A� 2 Dbc .Ey/, the derived stalks are

Hi .�cone

�p A�/x D

�

0 if x D c andi > p;

H i .A�/x otherwise.

260 FILIPP LEVIKOV

ForV 2 V , letSVf

be the local coefficient system overV with stalk.SVf/z D

H�c�t .EzIS�.z//. There is an induced nondegenerate bilinear pairing

�z W .SVf /z � .S

Vf /z! R:

If S� is the intersection chain complexIC �

Nm.X /, the pull-back�!y IC �

Nm.X / isclearly IC �

Nm.Ey n fcg/. Because of the stalk vanishing ofIC �

Nm.Ey n fcg/, thetruncation�cone

��c�t�1is the usual truncation��c�t�1 and hence

S�.y/D ��c�t�1Riy� IC �

Nm.Ey n fcg/;

which is simply the Deligne extensionIC �

Nm.Ey/ of IC �

Nm.Ey n fcg/ to the point.Denoting byIC �

Nm.NV ISV

f/ the lower-middle perversity intersection chain com-

plex on the closure ofV with coefficients in the local systemSVf

, we can nowformulate the important decomposition formula of Cappell and Shaneson.

THEOREM 4.5 [CS91, Theorem 4.2].There is an orthogonal decomposition upto algebraic bordism of self-dual complexes of sheaves

Rf�S�Œ�t ��M

V 2V

j� IC �

Nm.NV ;SV

f /Œc.V /�;

wherej W NV Œ Y is the inclusion.

We abstain from giving the definition of algebraic bordism here and refer tothe original paper or to Chapter 8 of [Ban07]. All we need for the applicationis the following, where for a self-dual sheafS� over X , �.X;S�/ denotes thesignature of the pairing on the middle-dimensional hypercohomology inducedby self-duality.

PROPOSITION4.6. If two self-dual complexes overX , S�

1and S�

2are (alge-

braically) bordant, then�.X;S�

1/D �.X;S�

2/.

PROOF. See [Ban07, Cor. 8.2.5], for example. ˜

PROPOSITION4.7 [CS91, 5.5].If , in the setting above, Li.X;A�/ denotes thei-th L-class of the self-dual sheafA� over a pseudomanifoldX , we have

Li.Y;Rf�S�Œ�t �/D f�Li.X;S�/:

THEOREM 4.8. With the notationLi. NV ;SVf/ for Li. NV ; IC �

Nm.NV ISV

f// we get

f�Li.X;S�/DX

V 2V

j�Li. NV ;SVf /:

and bearing in mind that�.X /D "�L0.X /, where"� is the augmentation, weconclude:

COROLLARY 4.9. �.X;S�/D �.Y;Rf�S�Œ�t �/DX

V 2V

�. NV ;SVf /:


In the case of simply connected components ofY , all the coefficient systemsbecome constant and using multiplicativity formulae [Ban07, 8.2.19, 8.2.20],we get:

THEOREM 4.10. Assume eachV 2 V to be simply connected and choose abasepointyV for everyV 2 V . Then

f�Li.X /DX

V 2V

�.EyV/j�Li. NV /:

And finally, for the signature:

COROLLARY 4.11. �.X /DX

V 2V

�.EyV/�. NV /:

5. The general simply connected case

PROPOSITION 5.1 (WANG SEQUENCE FORn � 2). Let F � E��! Sn

be a stratified bundle(2.1) with F a topological pseudomanifold with finitelygenerated cohomology, n� 2. Let j W F ŒE be the inclusion.

(i) For intersection homology the sequence

� � �� IHNp

k.E/

j�

�! IHNp

k�n.F /� IH

Np

k�1.F /

j�

�! IHNp

k�1.E/��

is exact.(ii) For intersection homology with compact supports the sequence

� � �� IHc Np

k.E/

j�

�! IHc Np

k�n.F /� IH

c Np

k�1.F /

j�

�! IHc Np

k�1.E/��

is exact.(iii) These sequences are natural with respect to fibre-preserving proper nor-

mally nonsingular maps between stratified bundles overSn, i.e., let F 0 !

E0! Sn be another fibre bundle such that there is a commutative triangle

Sn

E0 E........................................................................................

............

f

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.

.

.

.

.

.

.

..........

�

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

..........

.

.

.

.

.

.

.

.

.

.

.

.

� 0

with f proper normally nonsingular, then there is a commutative diagram ofthe corresponding Wang-sequences induced byf — both in a covariant anda contravariant way.

262 FILIPP LEVIKOV

PROOF. (i) We begin with the hypercohomology spectral sequence ([Bry93])for A� WD R�� IC �

Np.E/, which converges toHpCq.Sn;A�/ Š IHNp

�p�q.E/.

Let U � Sn be an open set such that��1.U /Š U �F , then by 4.3

IC �

Np.E/j��1.U / Š IC �

Np.U �F /Š pr� IC �

Np.F /Œn�:

By IV.7.3 of [Bre97], the sheafHq.A�/, being theLeray sheafof the fibration,is locally constant. Hence, by the assumptionn� 2, it is constant with stalk

Hq.F; IC �

Np.F /Œn�/D IH Np�q�n.F /:

Finally

Ep;q2Š

�

IHNp

�q�n.F / if p D 0 or p D n;

0 else.

HenceE2 Š : : :ŠEn and the sequence collapses atnC 1. Now, the proof canbe finished as in the ordinary case (see [Spa66, 8.5], for instance). In order toshow thatIH

Np

k.F /! IH

Np

k.E/ is induced by the inclusionj WF D b0�FŒE,

look at the fibration

F ! b0 �F� 0

�! b0

for b0 2 Sn the north pole. We have a commutative diagram

b0 �F b0

E Sn

........................................................................................................................................................................

............

� 0

............................................................................................................................................................................................

............

�

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

......

.

.

.

.

.

.

..

....

j

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

......

.

.

.

.

.

.

......

j0

:

For R.j0�0/� IC �

Np.b0�F / there is a corresponding spectral sequence converg-ing to

HpCq.b0 �F; IC �

Np.b0 �F //Š IH Np�p�q.F /:

If we start withR��Rj�j ! IC �

Np.E/!R�� IC �

Np.E/

and use the commutative square above, we get a morphism

R.j0�0/� IC �

Np.b0 �F /!R�� IC �

Np.E/:

This inducesIH

Npi .F /! IH

Npi .E/;

which is j� by construction (cf. 4.3 and 4.4). TheE2-term of the spectralsequence associated toR.j0�

0/� IC �

Np.b0 �F / is

Ep;q2DH

p.Sn;Hq.R.j0�0/� IC �

Np.b0 �F ///:


Since herej0� is just extension by zero the group on the right is isomorphicto

H p.b0; IHNp

�q.F //: For both sequences, the differentialsEn;qr ! E

nCr;q�rC1r

are zero for allr � 2, and so we have epimorphismsEn;qr “ E

n;q1 . Finally

by the commutative diagram (denoting by0 the terms of the spectral sequenceassociated toR.j0�

0/� IC �

Np.b0 �F /)

En;�i�nn E

n;�i�n1

E0n;�i�nn E

0n;�i�n1

Hq.E; IC �

Np.E//

Hq.F; IC �

Np.F //

.....................................................................................................................................

............

...........................................................................................................................

............

Š.....................................................................................

.

.

.

.

.

.

.

.....

ko.....................................................................................

.

.

.

.

.

.

.

.....

........................................................................................

............

.....................................................................................

............

Š........................................................................................

.

.

.

.

.

.

.

.....

j�

we deduce that the upper composition isj�, as stated. A very similar argumentworks forIH

Np

k.E/! IH

Np

k�n.F /.

(ii) Consider the hypercohomology spectral sequence for the complexB� WD

R�! IC �

Np.E/. The main argument is as before. The spectral sequence converges

toHpCq.Sn;B�/D IHc Np�p�q.E/. Being theqth derived functor of�!, the stalk

of the Leray sheafHq.B�/ is Hqc .F; IC

�

Np.E//Š IHc Np�q�n.F / (see [Bor84, VI,

2.7], for instance). Hence theE2-terms are:

E0;q2Š IH c Np

�q�n.F /;

En;q�nC12

Š IHc Np�q�1

.F /:

These yield the second sequence. The proof that the maps involved in this se-quence arej� andj � follows as in (i).

(iii) Again, we use the fact that the hypercohomology spectral sequence isnatural with respect to morphisms of sheaves over the base space.For the covariant case, we have to construct

f� WR�0

� IC �

Np.E0/!R�� IC �

Np.E/ or f� WR�0

! IC �

Np.E0/!R�! IC �

Np.E/;

as the case may be producing morphisms between the terms of the Wang se-quences. Then the corresponding maps will commute. Take theadjunctionmorphism

Rf!f! IC �

Np.E/! IC �

Np.E/;

apply R�� and use functoriality. The case ofIHc Np� is analogous. Sincef

is proper, we haveRf! D Rf�. Observe that when working with intersectionhomology with compact supportsf need not be proper4! In the contravariantcase, we proceed as above, using the other adjunction morphism

IC �

Np.E/!Rf�f� IC �

Np.E/: ˜

4See also the comment at the end of Section 4A.

264 FILIPP LEVIKOV

Now we are going to use this sequence in a concrete computation.

PROPOSITION5.2. Let Fk !E��! Sn be a locally trivial fibre bundle withF

a topological pseudomanifold, n � 2; nC kC 1 even. DefineM WD c E [ENE,

where NE is the total space of the induced — meaning that the structuregroupacts levelwise — fibre bundle

c F ! NEc .�/��! Sn:

Suppose further that the following condition(S) is fulfilled for the Wang se-quence ofE:

(S) the mapj � W IH c Nm.nCkC1/=2.E/“ IH c Nm

.�nCkC1/=2.F / is surjective.

ThenIH c Nm

.nCkC1/=2.M /D 0:

PROOF. Throughout the proof, the perversity shall be the lower middle perver-sity Nm, unless stated otherwise. Assume for now thatnD 2b, k D 2a� 1, witha; b � 1; and with the cone formula there holds

IH ci .cF /Š

�

IH ci .F / if i < a;

0 if i � a:

Using the Wang sequence of Proposition 5.1 forcF ! NEc.�/��! Sn,

� � � ! IH caCb.cF /! IH c

aCb.NE/! IH c

a�b.cF /! IH caCb�1.cF /! � � �

we get

IH caCb.

NE/Š IH ca�b.F /:

For the cone onE, there is:

IH ci .cE/Š

�

IH ci .E/if i < aC b;

0 if i � aC b:

Now consider the Mayer–Vietoris sequence5

� � � ! IH caCb.E/

iaCb

��! IH caCb.cE/˚ IH c

aCb.NE/! IH c

aCb.M /! � � �

which reduces to

� � �iaCb

��! IH ca�b.F /� IH c

aCb.M /�

IH caCb�1.E/

iaCb�1

��! IH caCb�1.E/˚ IH c

aCb�1.NE/��

5To avoid pseudomanifolds with boundary, we take the open part NE of the induced bundleNE in theMayer–Vietoris decomposition.


The mapiaCb�1 is easily seen to be injective and due to (S),iaCb is surjective.Finally, let nD 2bC 1; k D 2a; a; b � 1. By the cone formula we have:

IH ci .cE/Š

�

IH ci .E/ i < aC bC 1

0 i � aC bC 1

and

IH ci .cF /Š

�

IH ci .F / i < aC 1

0 i � aC 1:

Similarly, the Wang sequence yields

IH caCbC1.

NE/Š IH ca�b.F /:

Now, as above the Mayer–Vietoris sequence gives

iaCbC1��! IH c

a�b.F /! IH caCbC1.M /! IH c

aCb.E/iaCb�1��! IH c

aCb.E/˚IH caCb.

NE/

where keriaCb D 0 and imiaCbC1 D IH ca�b.F / due to (S). Hence,

IH caCbC1.M /D 0: ˜

COROLLARY 5.3. If M is a Witt space andnC k C 1 is divisible by4, thesignature�.M / vanishes(of course it always vanishes ifnCkC1 is not divisibleby 4.)

Let us now formulate an important consequence of the observations above:

PROPOSITION5.4. Let X be a Whitney stratified Witt space of dimension4k,with a disjoint union of spheres as the singular locus˙ D Sn1 t � � � t Snl .Assumenj � 2 for 1� j � l . LetY be the space obtained fromX by collapsingthe spheresSnj to pointsyj and letf W X ! Y be the collapsing map. Givena fibre bundle neighbourhood ofSnj , we denote byEj the corresponding fibrebundle with fibre the link ofSnj . If for all 1 � j � l , Ej satisfies(S), thesignature ofX does not change underf , i.e.,

�.X /D �.Y /:

PROOF. Because of 4.11, we have

�.X /DX

V 2V

�.EyV/�. NV /

where the sum is taken over all strata. When we isolate the contribution of thetop stratum, this looks like

DX

yj

�.Eyj V/�. NV /C �.Y /:

266 FILIPP LEVIKOV

TheEyj V, in turn, are of the formM of Proposition 5.2 and since the underlying

fibrations satisfy (S), the resulting signatures vanish by 5.3. ˜

Similarly for theL-classes we have:

PROPOSITION5.5. In the situation above,

f�L.X /DL.Y /:

The following two examples show, that the introduced condition (S) is indeedfulfilled for certain fibre bundles.

EXAMPLE 5.6. In the setting above let the base sphere be of odd dimensionnD 2bC1. If we are interested in computing the signature ofE, its dimensionhas to be divisible by 4 — otherwise it is trivial anyway. In this case the fibreFhas even dimensionk D 2a, so that.k � nC 1/=2 is odd. Thus, the vanishingof odd dimensional intersection homology ofF would imply (S). See [Roy87]for examples of spaces, for which the intersection homologyvanishes in odddegrees.

EXAMPLE 5.7. Let the dimension of the sphere be greater than the dimensionof the fibre plus 1, i.e.,k C 1 < n. Then.k � nC 1/=2 is negative and thecorresponding homology group is zero, thereby (S) is fulfilled.

Since we have not studied the intersection pairing onM , the condition (S) isclearly only sufficient and not necessary. However, the following ”counterex-ample” to the proposition is a case where (S) does not hold.

EXAMPLE 5.8. LetX beCP 2 stratified asCP 2�CP 1DS2 andf be the map,collapsing the 2-sphere to a point. So the target isY D S4 � ŒS2�. Obviously,�.X /¤ �.Y /. The link ofCP 1 is a circle and the bundle we have to check (S)for is the Hopf bundleS3! S2. However

H2.S3/!H0.S

2/

is not onto and (S) fails.

6. A new proof

The application to the signature in the last section suggests a similar approachin the setting of spaces which no longer satisfy the Witt condition, howeverstill posses a signature andL-classes. The suitable homology groups for defin-ing these invariants are the hypercohomology groupsHi.�I IC �

L/ of Banagl[Ban02]. In the next section we will establish a Wang-like exact sequence forthese groups i.e., for hypercohomology with values in a self-dual sheaf complexarising from a Lagrangian structure along the odd-codimension strata. Compare


also [Ban07] for a concise exposition. The proof will be modeled on anotherelegant proof of the Wang sequence without the usage of the spectral sequence.This will be demonstrated in the following.

Suspension isomorphisms in intersection homology are veryfamiliar. Forthe functoriality, however, we would like to have explicit maps realizing theseisomorphisms:

LEMMA 6.1 (SUSPENSION ISOMORPHISM). Let F be a pseudomanifold. Theinclusionl W F D 0�F Œ Rn �F induces isomorphisms

(a) l� W IHNp

k.Rn �F /! IH

Np

k�n.F /,

(b) l! W IHc Np

k.F /! IH

c Np

k.Rn �F /.

PROOF. (a) Letp W Rn � F ! F be the normally nonsingular projection. By[Bor84, V,3.13]Rp� ıp� ' id, so the adjunction morphism is an isomorphism

IC �

Np.F /Š

�!Rp�p� IC �

Np.F /ŠRp� IC �

Np.Rn �F /Œ�n�:

Applying hypercohomology we get

p� W IHNp

k.F /

Š

�! IHNp

kCn.Rn �F /:

Now p ı l D id and hencel� ıp�' id. Thereby,l� is the inverse ofp� and thestatement follows.

(b) is similar to (a), but uses the fact thatDX DX A� Š A� for A� 2Dbc .X /, and

the duality betweenp� andp!. ˜

PROPOSITION6.2. Let F � E��! Sn be a stratified bundle withF a topo-

logical pseudomanifold. Denote by

j W F D b0 �F ŒE, i WE n b0 �F D U �F ŒE, k W b1 �F ŒE

the inclusions, whereb0 is the north pole andb1 the south pole. Then there arethe following long exact sequences:

� � �� IHNp

k.F /

j�

�! IHNp

k.E/

k�

��! IHNp

k�n.F /� IH

Np

k�1.F /��

� � �� IHc Np

k.F /

k!�! IH

c Np

k.E/

j�

�! IHNp

k�n.F /� IH

c Np

k�1.F /�� :

PROOF. In the following, trivializations of the fibre bundleE are always in-volved. However, for every pseudomanifoldX , h WX Š�!X implies IC �

Np.X /Š

h� IC �

Np.X /Š h� IC �

Np.X /. Therefore, for the proof we can suppress them.

268 FILIPP LEVIKOV

We begin with the distinguished triangle

Rj�j ! IC �

Np.E/ IC �

Np.E/

Ri�i� IC �

Np.E/

...........................................................................................................................

............

...................................................................................................................

............

...................................................................................................................

............

Œ1�

and keeping in mind thatj ! IC �

Np.E/Š IC �

Np.F /, i� IC �

Np.E/Š IC �

Np.U �F / weapply hypercohomology to get

� � � ! IHNp

k.F /

j�

�! IHNp

k.E/

i�

�! IHNp

k.U �F /! � � � :

The third term is isomorphic toIHNp

k�n.F / underl� by the preceding lemma.

However by the commutative triangle

b1 �F E

U �F

.................................................................................................................................................................................................

............

k

.....................................................................................................................

............

i

..............................................................................................................

............l

we havek� ' l� ı i� and the sequence for closed supports is proven.

Now turn to the case of compact supports.6 Consider the triangle

Ri!i� IC �

Np.E/ IC �

Np.E/

Rj�j � IC �

Np.E/

................................................................................................................................

............

...................................................................................................................

.

...........

.......................................................................................................................

............

Œ1�

and apply hypercohomology with compact supports to get

� � �!H�kc .EIRi! IC �

Np.U�F //i!�!IH

c Np

k.E/

j�

�!H�kc .EIRj� IC �

Np.F /Œn�/!� � � :

Now Rj� DRj! asj is a closed inclusion. Hence

H�kc .EIRj� IC �

Np.F /Œn�/ŠH�kc .F I IC�

Np.F /Œn�/Š IHc Np

k�n.F /:

For the first term, we have

H�kc .EIRi! IC �

Np.U �F //ŠH�kc .U �F I IC �

Np.U �F //l! �Š

IHc Np

k.F /

and withi! ı l! D k! the assertion follows. ˜

6Recall that forf W X ! Y , A� 2 Dbc .X / andZ � X , we have�c.Z; f�A�/©�c.f

�1.Z/;A�/.However�c.Z; f!A�/Š �c.f

�1.Z/;A�/.


7. The non-Witt case

7A. The category SD(X ). Originally, Goresky and MacPherson defined thesignature for spaces with only even-codimensional strata.In [Sie83], Siegelgeneralizes the definition to Witt spaces. IfIH Nm

�.X /© IH Nn

�.X /, there still is a

method to define a signature andL-classes for a pseudomanifoldX compatiblewith the old definition. In his work [Ban02], Banagl establishes a correspondingframework and decomposition results similar to those of Cappell and Shanesonare presented in further papers. In this section we merely give the definition.

DEFINITION 7.1. LetX DXn � � � � �X0 be an oriented pseudomanifold withorientation

o W D�

U2

Š

�! RU2Œn�:

For k � 2, we writeUk WDXnXn�k . DefineSD.X / as the full subcategory ofDb

c .X / of thoseS� 2Dbc .X / satisfying the following:

(SD1) Normalization: There is an isomorphism� W RU2Œn�

Š

�! S�jU2.

(SD2) Lower bound:Hi.S�/D 0, for i < �n.(SD3) Stalk condition forNn: Hi.S�jUkC1

/D 0, for i > Nn.k/� n; k � 2.(SD4) Self-duality: There is an isomorphismd WDX S�Œn�!S� compatible with

the orientation, i.e., such that the square

RU2Œn� S�jU2

D�

U2DX S�jU2

Œn� commutes.

....................................................................................................................................................................................

............

�...................................................................................................................................................................................

.

............

Š

...................................................................................................................................................................

............

DX ��1Œn�

...................................................................................................................................................................

............

Š

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

o

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

Š

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

d jU2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

Š

We refer to [Ban02] for results on this category, especiallyfor the structuretheorem, establishing the relation betweenS� 2 SD.X / and a choice of La-grangian structures along odd-codimensional strata ofX .

REMARK 7.2. If X is a Witt space,SD.X / consists up to isomorphism only ofIC �

Nm.X /. On the other hand,SD.X /might be empty — e.g.,SD.˙CP 2/D?.

THEOREM7.3 [Ban02, Theorem.2.2].For S� 2SD.X /, there is a factorization

IC �

Nm.X /˛�! S�

ˇ�! IC �

Nn.X /;

270 FILIPP LEVIKOV

that is compatible with the normalization(and is unique with respect to thisproperty) and such that

IC �

Nm.X / S�

DX IC �

Nn.X / DX S� commutes.

.........................................................................................................................................................................................

............

˛

......................................................................................................................................................

............

DX ˇŒn�.............................................................................................................................

.

.

.

.

.

.

.

.....

Š

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

d

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

Š

Thus, an object inSD.X / is in fact a self-dual interpolation betweenIC �

Nm.X /

and IC �

Nn.X /. It is obvious that in the case ofX being a Witt space,SD.X /

consists (up to quasi-isomorphism) only ofIC �

Nm.X /.

DEFINITION 7.4. LetX n be a closed stratified topological pseudomanifold, notnecessarily Witt andS�2SD.X /. In casen is divisible by 4, define�.X n;S�; d/

to be the signature onH�n=2.X n;S�/ induced by the self-duality ofS�.

REMARK 7.5. If X happens to be a Witt space,�.X n;S�; d/ is the usual inter-section homology signature due to Theorem 7.3.

Finally, in order to speak ofthe signature of a pseudomanifold (as long asSD.X /¤?), we need the following important result:

THEOREM 7.6 [Ban06, 4.1].Let X n be an even-dimensional closed orientedpseudomanifold withSD.X /¤?. For .S�

1; d1/; .S�

2; d2/ 2 SD.X / one has

�.X n;S�

1; d1/D �.Xn;S�

2; d2/:

7B. Hypercohomology Wang sequence.Before we deduce the exact sequencefor hypercohomology with values inSD-sheaves, we have to determine whatthe involved complexes of sheaves are going to be. Starting with a SD complexover the total spaceE we define a SD complex over the fibreF in a canonicalway. We will need the following little lemma.

LEMMA 7.7. For the inclusionj WX n�0ŒX n�Rm with associated projectionp WX n �Rm! X n we have

j ! ' j �Œ�m�

and therebyp! ı j ! ' id :

PROOF. By [Ban02, Lemma 5.2],p� ı j � ' id. Consequently, usingp�Œm�'

p!([Ban02, Lemma 4.2, Proof]), we get

j ! ' j ! ıp� ı j � ' j ! ıp! ı j �Œ�m�' .p ı j /! ı j �Œ�m�' j �Œ�m�:


The second identity is clear by usingp�Œm�' p! again. ˜

LEMMA 7.8. Let X nj�! X n � Rm be the standard inclusion. Given T� 2

SD.X n �Rm/, the complexj !T� is in SD.X /.

PROOF. Looking at the commutative square

RU2ŒnCm� T�jU2

j !RU2ŒnCm�Š RU2\X Œn� j !T�jU2\X

........................................................................................................................................................................................................................................................................................

............

�

...................................................................................................................................................................

............

j !.�/

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

j !

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

j !

;

one checks that (SD1) is fulfilled because of the functoriality of j !. Now usingthat the inverse image functorj � is exact, look at

Hi.j !T�/Š Hi.j �T�Œ�m�/Š Hi�m.j �T�/Š j � Hi�m.T�/:

Observe that the last term is zero fori�m<�.nCm/ or i <�n, and so (SD2)holds. Now leti > Nn.k/� .n/; k � 2. We have

Hi..j !T�/jUkC1\X /Š Hi..j �T�Œ�m�/jUkC1\X /Š j � Hi�m.T�jUkC1/;

where the last term is zero due toi�m> Nn.k/�.nCm/ and hence (SD3) holdsas well.Finally by [Ban07, Proposition 3.4.5] we have an isomorphism

DX j !T�Œn�Š j �DX �RmT�Œn�Š j !.DX �RmT�ŒnCm�/Š j !T�

which is compatible with the orientation, proving (SD4). ˜

Let us now return to the original context. We start with the total spaceE ofa fibre bundle overSn — a topological pseudomanifold of dimensionkC n —and a complexT� 2SD.E/. Given a trivializing neighbourhoodU �Sn of thenorth poleb0 resp. the south poleb1, the restriction ofT� to ��1.U /Š U �F

is clearly inSD.U �F / since��1.U / is open. With the preceding lemma wecan now define:

DEFINITION 7.9. LetF!E!Sn be a fibre bundle as above andT�2SD.E/.For i D 0; 1, we have inclusions

bi �F E

U �F ��1.U /

...................................................................................................................................................................................................

............

ji

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

i2i

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

i1i

........................................................................................................................................................................

............

�i

........................................................................................................................................................................

............

Š

272 FILIPP LEVIKOV

whereji is defined to be the compositioni2i ı�i ı i1

i .SetS� WDS�

NWD j !

0T�Š i1!

0..�!

0T�/jU �F / 2 SD.F /, with j0 W b0�FŒ the

inclusion of the north pole fibre for some trivialization�0 WU �FŠ

�! ��1.U /.DefineS�

Sin the same way using the inclusion of the south pole fibre.

In order forS� to be well defined overF D bi �F with bi 2 U1\U2, we haveto make an extra assumption, a kind of homogeneity:

DEFINITION 7.10. We call the structure groupG of a fibre bundle of the formaboveadapted toA� 2 SD.F /, if for all h 2G, h!A� Š h�A� Š A�.

EXAMPLE 7.11. If F is a Witt space,IC �

Nm.F / 2 SD.F /. For every stratum-

preserving automorphismh W FŠ

�! F , we haveh� IC �

Nm.F /Š IC �

Nm.F /.

REMARK 7.12. AssumeG to be adapted toS�. What if we are given twotrivializing neighbourhoodsU1;U2 � Sn with �i W Ui � F ! ��1.Ui/? Wehave a commutative diagram

F b0 �F

F b0 �F

.U1\U2/�F

.U1\U2/�F

��1.U1\U2/ E

.............................................................................................

............

i2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

Š

.............................................................................................

............

i1...............................................................................

.

............

................................................................................

............

.....................................................................................

............

�2

.....................................................................................

............

�1

...............................................................

............

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.....

.

.

.

.

.

.

.

.....

Š........................................................................................

.

.

.

.

.

.

.

.....

h21.b0/........................................................................................

.

.

.

.

.

.

.

.....

��12�1

where the upper composition isj 1 and the lower composition isj 2. We have toshow that.j 1/!T� D .j 2/!T�. SinceG is adapted toS�, the transition functionh21.b0/ preservesS� and thereby.j 1/!T� D h21.b0/

!.j 2/!T� Š .j 2/!T�.

We will need some form of suspension isomorphism for hypercohomology withvalues in a SD sheaf.

LEMMA 7.13.Let F be a pseudomanifold andS� 2SD.F /. Letp WRn�F!F

be the projection. The inclusionl W 0 � F Œ Rn � F induces the followingisomorphisms on hypercohomology:

(a) l� WHk.Rn �F IS�/!HkCn.F I l !S�/

(b) l! WHkc .F I l

!S�/!Hkc .R

n �F IS�/

PROOF. Same as for Lemma 6.1, usingp� ' p!Œ�n� for (a). Note that by[Bor84, V, 3.13]Rp�p�A� Š A� for all A� 2Db.X /. ˜

Now we are able to formulate the next proposition and imitatethe proof of theWang sequence given in Section 6.

PROPOSITION7.14.Let F !E! Sn be a fibre bundle with a suitable struc-ture group G of automorphisms of the pseudomanifoldF , which is adapted to


S�

NandS�

S. AssumeE to be canonically stratified. GivenT� 2 SD.E/ there

are long exact sequences

(a) � � � !Hk.F IS�/!Hk.EIT�/!HkCn.F IS�/! � � �,

(b) � � � !Hkc .F IS

�/!Hkc .EIT

�/!HkCnc .F IS�/! � � �,

whereS� is the self-dual sheaf of Definition7.9.

PROOF. Due to Remark 7.12 we need not pay attention to different trivializa-tions. Therefore, in the following proof we will not mentionthem explicitly.

(a) Denote byj the inclusion of the north pole fibre and byi the inclusion ofthe complementV �F . We begin with the distinguished triangle

Rj�j !T� T�

Ri�i�T�

.......................................................................................................................................................................................

............

.....................................................................................................................................

.

.............

.................................................................................................................

............

Œ1�

and apply hypercohomology to get

� � � !Hk.F I j !T�/!H

k.EIT�/!Hk.V �F I i�T�/! � � � :

By construction, we haveHk.F I j !T�/ŠHk.F IS�/. If i WV �F!E denotesthe inclusion, theni�T� is again isomorphic to a self-dual complex over theproduct bundleV �F containing the south pole fibreb1�F . With the inclusionj1 W b1 �F ŒE and using the preceding lemma, we finally get

Hk.V �F I i�T�/DH

kCn.F I j !1T�/ŠH

kCn.F IS�

S /:

Now choose a trivializing neighbourhoodO containing the north poleb0 andthe south poleb1. Let l0 andl1 be the corresponding inclusions intoO�F . Wehave

S� D j !0T� D l !

0.T�jO�F /Š l !

1.T�jO�F /Š j !

1T� D S�

S ;

where the middle isomorphism holds becauseG is adapted toS�

NandS�

S. This

completes the proof.

(b) Begin with the triangle

Ri!i�T� T�

Rj�j �T�

.........................................................................................................................................................................................

............

.................................................................................................................................

............

.......................................................................................................................

............

Œ1�

and apply hypercohomology with compact supports to get

� � � !Hkc .V �FI i�T�/!H

kc .EIT

�/!Hkc .FIj

�T�/! � � � :

274 FILIPP LEVIKOV

Now, using part (b) of the preceding lemma, we identify

Hkc .V �F I i�T�/ŠH

kc .F I j

!1i�T�/ŠH

kc .F IS

�/:

And again by using

b0 �F E

U �F

.................................................................................................................................................................................................

............

j

........................................................................................................

.

...........

i2

.......................................................................................................

............i1

we obtain

j �T� Š i�

1 i�

2 T� Š i !1.T

�jV �F /Œn�Š S�Œn�:

Hence the third term in the sequence is equal toHkc .F I j

�T�/ŠHkCnc .F IS�/.

The observationS� Š S�

Sholds as before. ˜

REMARK 7.15. We can still formulate a similar exact sequence even withoutGbeing adapted toS�

NandS�

Sor using the local triviality of the stratifold bundle

only. Then, however, the involvedSD-complexes over the fibre may be differentand depend on the choice of trivializations:

� � � !Hk.F IS�

N /!Hk.EIT�/!H

kCn.F IS�

S /! � � � ;

� � � !Hkc .F IS

�

S /!Hkc .EIT

�/!HkCnc .F IS�

N /! � � � :

8. Novikov additivity and collapsing of spheres

In [Sie83], Siegel generalizes the classical Novikov additivity of the signaturefor manifolds to pseudomanifolds satisfying the Witt condition. We want tomake a further step forward by dropping the Witt condition incertain cases.7

PROPOSITION8.1. Let X DX2n �X2n�k �?, k � 2, be a Whitney stratifiedcompact pseudomanifold withSD.X / ¤ ?. Given subspacesM;E;T � X

with T a closed neighbourhood ofX2n�k , such that

(1) X DM [T ,(2) M \T DE,(3) E has a collar inT , and(4) .M;E/ is a compact manifold with boundary.

DefineX 1 WDM [E c.E/ andX 2 WD T [E c.E/. Then we have the identity

�.X /D �.X 1/C �.X 2/:

7A similar result is given in Theorem 3 of [Hun07]. Thanks to the referee for pointing this out to me.


PROOF. Once again, consider the formula of Cappell and Shaneson 4.5. Letf W X ! Y be the map collapsingX2n�k to a point. The push forwardRf�T�

of T� 2 SD.E/ is algebraically cobordant to

IC �

Nm.X1;SM

f /˚ jc� IC �

Nm.fcg;Sfcg

f/Œn�:

Here,c is the conepointf .X2n�k/ andjc its inclusion intoY . SinceSMf

isconstant with rank one, the first term is just equal toIC �

Nm.X1/. Let us look at

.Sfcg

f/c. The linkL.c/ of c is E, so we deduce from

Ec D f�1.c L.c//[E cf �1.L.c//DX 2

andEc

ic

‹Ecnfcg Š f�1.c L.c//

�c

ŒX

(see Section 4B) thatS�.c/ is a Deligne extension of�!cS� — which is just the

restriction of the original self-dual complex. Hence8 S�.c/2SD.X2/. We have.S

fcg

f/c DH�n.X 2IS�.c// and consequently

H�n.Y I jc� IC �

Nm.fcg;Sfcg

f/Œn�/ŠH

�n.fcgI IC �

Nm.fcg;Sfcg

f/Œn�/

ŠH�n.X 2IS�.c//:

Finally, combining these observations and passing to the signature we get

�.X /D �.X;Rf�.T�//

D �.X 1; IC �

Nm.X1//C �.X 2;S�.c//D �.X 1/C �.X 2/: ˜

Let E be the total space of a fibre bundle overSm as before. We investigate,when the middle hypercohomology groupH�n.M;S�/ vanishes for a givencomplexS� over a non-WittM WD c E [E

NE of dimension2n. Since we areonly interested in computing the signature, only odd-dimensional spheres areconsidered here. The strategy is very similar to that of Section 5.

PROPOSITION8.2. LetF2a!E��!S2bC1 be a fibre bundle withF a compact

topological pseudomanifold, a� 1, b � 2. DefineM WD c E [ENE, where NE is

the total space of the induced fibre bundle

c F ! NEc .�/��! S2bC1:

Given S� 2 SD.M /, denote byT� the induced element inSD.E/ (compareSection7). Assume that the following condition(S) is fulfilled for the hyper-cohomology Wang sequence ofE:

H�.aCbC1/.EIT�/“ H

�.a�b/.F ISS�/ is surjective,

8All the axioms are clearly satisfied. The only “new” stalk to look at is the one atc, but fcg has evencodimension and the modified Deligne extension of�!

cS� explained in 4B ensures that SD1-SD4 remain valid.

276 FILIPP LEVIKOV

Then

H�.aCbC1/.M IS�/D 0:

We will need the following vanishing lemma for the hypercohomology of thecone.

LEMMA 8.3. Let X be a2n-dimensional or a.2nC 1/-dimensional, compactWitt space such that the signature of the pairing over the middle-dimensionalintersection homology vanishes. For S� 2 SD.cX /, we have

H�ic .cX IS�/D 0 for i � nC 1:

PROOF. SinceS� is constructible andcX is a distinguished neighbourhood ofthe conepointc, there is (by [Ban07, p. 97], for instance)

H�ic .cX IS�/DH

�i.j !cS�/

where the latter is the costalk ofS� at c. Because of self-duality, however,S�

satisfies the costalk vanishing condition

H�i.j !

cS�/D 0 for � i �

�

Nm.2nC 1/�dim cX C 1D�.nC 1/;

Nm.2nC 2/�dim cX C 1D�.nC 1/;

which is equivalent to the statement. ˜

PROOF OFPROPOSITION8.2.. Look at the hypercohomology Wang sequence

for cF ! NE! S2bC1

� � � !H�.aCbC1/c .cF IU�/!H

�.aCbC1/c . NEIT�/!H

�.a�b/c .cF IU�/! � � �

whereT�DS�jNE

is in SD. NE/ andU� 2SD.cF / is constructed as in 7.9. Sinceb � 2, we see from the preceding lemma that

H�.aCbC1/c .cF IU�/DH

�.aCb/c .cF IU�/D 0

and hence

H�.aCbC1/c . NEIT�/ŠH

�.a�b/c .cF IU�/:

DecomposeM into the open subsetsNE and cE with NE \ cE D E � .0; 1/.Consider the Mayer–Vietoris hypercohomology sequence (see [Ive86, III.7.5]or [Bre97, II,~ 13], for instance)

� � ��H�.aCbC1/c .E � .0; 1/IS�j/

iaCbC1

��!

H�.aCbC1/c .cEIS�j/˚H

�.aCbC1/c . NEIS�j/�

H�.aCbC1/c .M IS�/�H

�.aCb/c .E � .0; 1/IS�j/

iaCb

��! � � �


Let us first show thatiaCb is injective. We use the following exact sequence forhypercohomology with compact supports (see [Ive86, III.7.7])

� � �!H�.aCbC1/c .fcgIS�/!H

�.aCb/c .cEnfcgIS�/!H

�.aCb/c .cEIS�/!� � � :

It suffices to show that

H�.aCbC1/c .fcgIS�/ŠH

�.aCbC1/.fcgIS�/

vanishes. The latter is isomorphic toH�.aCbC1/.S�

c/ which is 0 because of

(SD3). Due to the preceding lemmaH�.aCbC1/c .cEIS�j/ is 0. The surjectivity

of iaCbC1 follows now from the condition (S) using the naturality of the hyper-cohomology Wang sequence in completely the same manner as inthe proof ofProposition 5.2. ˜

REMARK 8.4. As you can see, we have only used the vanishing lemma 8.3,which is valid for everyS� 2 SD.F /. Hence, in the proposition, the structuregroupG need not be adapted.

REMARK 8.5. In the case of a Witt fibreF the condition (S) is the one ofSection 5. See the examples there, especially 5.6.

COROLLARY 8.6. Let M be as in8.2. The signature�.M / of M vanishes.

COROLLARY 8.7. Let X be as in8.1 such thatXn�k is an odd-dimensionalsphere. AssumeE to satisfy(S). Then�.X /D �.X 1/D �.M; @M /, where thelatter is the Novikov signature ofM .

Acknowledgements

The article is based on the author’s Diploma thesis at the University of Hei-delberg [Lev07] and is motivated by the MSRI workshop on the Topology ofStratified Spaces. I want to express my gratitude to Greg Friedman, EugenieHunsicker, Anatoly Libgober and Laurentiu-George Maxim for the organisationof this event, the accompanying support and the opportunityfor giving a talk.I am also indebted to my thesis advisor Markus Banagl for his help and adviceduring the development of the thesis. Last but not least I am grateful to thereferee for the careful reading of the preliminary version.

References

[Ban02] M.Banagl,Extending intersection homology type invariants to non-Witt spaces,Mem. Amer. Math. Soc.160(2002), no. 760.

[Ban06] , The L-class of non-Witt spaces, Ann. of Math.163(2006), no. 3.

278 FILIPP LEVIKOV

[Ban07] , Topological invariants of stratified spaces, Springer Monographs inMathematics, Springer, 2007.

[Bor84] A. Borel et al.,Intersection cohomology, Progr. Math., vol. 50, Birkhauser,Boston, 1984.

[Bre97] G. E. Bredon,Sheaf theory, second ed., Grad. Texts in Math., vol. 170,Springer, New York, 1997.

[Bry93] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantiza-tion, Progr. Math., vol. 107, Birkhauser, Boston, 1993.

[CS91] S. E. Cappell and J. Shaneson,Stratifiable maps and topological invariants, J.Amer. Math. Soc.4 (1991), 521–551.

[FM81] W. Fulton and R. MacPherson,Categorical framework for the study of singularspaces, Mem. Amer. Math. Soc.31 (1981), no. 243.

[Fri07] G. Friedman,Intersection homology of stratified fibrations and neighborhoods,Adv. Math.215(2007), 24–65.

[GM83] M. Goresky and R. Macpherson,Intersection homology II, Invent. Math.72(1983), no. 1, 77–129.

[GM88] M. Goresky and R. MacPherson,Stratified morse theory, Ergebnisse derMathematischen Wissenschaften, vol. 14, Springer, BerlinHeidelberg, 1988.

[HS91] N. Habegger and L. Saper,Intersection cohomology of cs-spaces and Zeeman’sfiltration, Invent. Math.105(1991), no. 2, 247–272.

[Hud69] J. F. P. Hudson,Piecewise linear topology, Mathematics Lecture Notes Series,W. A. Benjamin, New York, 1969.

[Hun07] E. Hunsicker,Hodge and signature theorems for a family of manifolds withfibration boundary, Geom. Top.11 (2007), 1581–1622.

[Ive86] B. Iverson,Cohomology of sheaves, Universitext, Springer, Berlin, 1986.

[Kin85] H. King, Topological invariance of intersection homology without sheaves,Topology Appl.20 (1985), 149–160.

[Lev07] F. Levikov,Wang sequences in intersection homology, Diploma Thesis, Uni-versitat Heidelberg, 2007.

[Roy87] J. Roy,Vanishing of odd-dimensional intersection cohomology, Math. Z.195(1987), 239–253.

[Sie83] P.H. Siegel,Witt spaces: A geometric cycle theory forKO-homology at oddprimes, American J. Math.105(1983), 1067–1105.

[Spa66] E. Spanier,Algebraic topology, McGraw-Hill, New York, 1966.

[Wan49] H. C. Wang,The homology groups of fibre bundles over a sphere, Duke Math.J.16 (1949), 33–38.


FILIPP LEVIKOV

INSTITUTE OFMATHEMATICS

K ING’ S COLLEGE

ABERDEEN AB24 3UESCOTLAND

[email protected]


An exponential history of functions withlogarithmic growth

MATT KERR AND GREGORY PEARLSTEIN

ABSTRACT. We survey recent work on normal functions, including limits andsingularities of admissible normal functions, the Griffiths–Green approach tothe Hodge conjecture, algebraicity of the zero locus of a normal function,Neron models, and Mumford–Tate groups. Some of the material and manyof the examples, especially in Sections 5 and 6, are original.

Introduction

CONTENTS

Introduction 2811. Prehistory and classical results 2852. Limits and singularities of normal functions 3003. Normal functions and the Hodge conjecture 3144. Zeroes of normal functions 3265. The Neron model and obstructions to singularities 3446. Global considerations: monodromy of normal functions 359References 368

In a talk on the theory of motives, A. A. Beilinson remarked that according tohis time-line of results, advances in the (relatively young) field were apparently alogarithmic function oft ; hence, one could expect to wait 100 years for the nextsignificant milestone. Here we allow ourselves to be more optimistic: followingon a drawn-out history which begins with Poincare, Lefschetz, and Hodge, thetheory ofnormal functionsreached maturity in the programs of Bloch, Griffiths,

The second author was supported by NSF grant DMS-0703956.

281

282 MATT KERR AND GREGORY PEARLSTEIN

Zucker, and others. But the recent blizzard of results and ideas, inspired byworks of M. Saito on admissible normal functions, and Green and Griffiths onthe Hodge Conjecture, has been impressive indeed. Besides further papers oftheirs, significant progress has been made in work of P. Brosnan, F. Charles,H. Clemens, H. Fang, J. Lewis, R. Thomas, Z. Nie, C. Schnell, C. Voisin,A. Young, and the authors — much of this in the last 4 years. This seems like agood time to try to summarize the state of the art and speculate about the future,barring (say) 100 more results between the time of writing and the publicationof this volume.

In the classical algebraic geometry of curves, Abel’s theorem and Jacobiinversion articulate the relationship (involving rational integrals) between con-figurations of points with integer multiplicities, or zero-cycles, and an abelianvariety known as the Jacobian of the curve: the latter algebraically parametrizesthe cycles of degree 0 modulo the subgroup arising as divisors of meromorphicfunctions. Given a familyX of algebraic curves over a complete base curveS , with smooth fibers overS� (S minus a finite point set over which fibershave double point singularities), Poincare [P1; P2] definednormal functionsasholomorphic sections of the corresponding family of Jacobians overS whichbehave normally (or logarithmically) in some sense near theboundary. His mainresult, which says essentially that they parametrize1-dimensional cycles onX ,was then used by Lefschetz (in the context whereX is a pencil of hyperplanesections of a projective algebraic surface) to prove his famous .1; 1/ theoremfor algebraic surfaces [L]. This later became the basis for the Hodge conjecture,which says that certaintopological-analyticinvariants of analgebraicvarietymust come fromalgebraicsubvarieties:

CONJECTURE1. For a smooth projective complex algebraic varietyX , withHgm.X /Q the classes inH 2m

sing.XanC ;Q/ of type.m;m/, andCH m.X / the Chow

group of codimension-m algebraic cycles modulo rational equivalence, the fun-damental class mapCH m.X /˝Q! Hgm.X /Q is surjective.

Together with a desire to learn more about the structure of Chow groups (theBloch–Beilinson conjectures reviewed in~ 5), this can be seen as the primarymotivation behind all the work described (as well as the new results) in thispaper. In particular, in~ 1 (after mathematically fleshing out the Poincare–Lefschetz story) we describe the attempts to directly generalize Lefschetz’ssuccess to higher-codimension cycles which led to Griffiths’ Abel–Jacobi map(from the codimensionm cycle group of a varietyX to its m-th “intermedi-ate” Jacobian), horizontality and variations of mixed Hodge structure, and S.Zucker’s Theorem on Normal Functions. As is well-known, thebreakdown(beyond codimension 1) of the relationship between cycles and (intermediate)

AN EXPONENTIAL HISTORY OF FUNCTIONS WITH LOGARITHMIC GROWTH 283

Jacobians, and the failure of the Jacobians to be algebraic,meant that the samegame played in 1 parameter would not work outside very special cases.

It has taken nearly three decades to develop the technical underpinnings fora study of normal functions over ahigher-dimensional baseS : Kashiwara’swork on admissible variations of mixed Hodge structure [K],M. Saito’s intro-duction of mixed Hodge modules [S4], multivariable nilpotent and SL2-orbittheorems ([KNU1],[Pe2]), and so on. And then in 2006, Griffiths and Greenhad a fundamental idea tying the Hodge conjecture to the presence ofnontor-sion singularities— nontrivial invariants in local intersection cohomology —for multiparameter normal functions arising from Hodge classes on algebraicvarieties [GG]. We describe their main result and the follow-up work [BFNP]in ~ 3. Prior to that the reader will need some familiarity with the boundarybehavior of “admissible” normal functions arising from higher codimension al-gebraic cycles. The two principal invariants of this behavior are calledlimitsandsingularities, and we have tried in~ 2 to give the reader a geometric feel forthese through several examples and an explanation of the precise sense in whichthe limit of Abel–Jacobi invariants (for a family of cycles)is again some kind ofAbel–Jacobi invariant. In general throughout~ ~ 1–2 (and~ 4.5–6) normal func-tions are “of geometric origin” (arise from cycles), whereas in the remainder theformal Hodge-theoretic point of view dominates (though Conjecture 1 is alwaysin the background). We should emphasize that the first two sections are intendedfor a broad audience, while the last four are of a more specialized nature; onemight say that the difficulty level increases exponentially.

The transcendental (nonalgebraic) nature of intermediateJacobians meansthat even for a normal function of geometric origin, algebraicity of its vanishinglocus (as a subset of the baseS), let alone its sensitivity to the field of def-inition of the cycle, is not a foreordained conclusion. Following a review ofSchmid’s nilpotent and SL2-orbit theorems (which lie at the heart of the limitmixed Hodge structures introduced in~ 2), in ~ 4 we explain how generaliza-tions of those theorems to mixed Hodge structures (and multiple parameters)have allowed complex algebraicity to be proved for the zero loci of “abstract”admissible normal functions [BP1; BP2; BP3; S5]. We then address the fieldof definition in the geometric case, in particular the recentresult of Charles[Ch] under a hypothesis on the VHS underlying the zero locus,the situationwhen the family of cycles is algebraically equivalent to zero, and what all thismeans for filtrations on Chow groups. Another reason one would want the zerolocus to be algebraic is that the Griffiths–Green normal function attached to anontrivial Hodge class can then be shown, by an observation of C. Schnell, tohave a singularity in the intersection of the zero locus withthe boundary �S

(though this intersection could very well be empty).


Now, a priori, admissible normal functions (ANFs) are only horizontal andholomorphic sections of a Jacobian bundle overSn˙ which are highly con-strained along the boundary. Another route (besides orbit theorems) that leads toalgebraicity of their zero loci is the construction of a “Neron model” — a partialcompactification of the Jacobian bundle satisfying a Hausdorff property (thoughnot a complex analytic space in general) and graphing admissible normal func-tions over all ofS . Neron models are taken up in~ 5; as they are better un-derstood they may become useful in defining global invariants of (one or more)normal functions. However, unless the underlying variation of Hodge structure(VHS) is a nilpotent orbit the group of components of the Neron model (i.e., thepossible singularities of ANFs at that point) over a codimension� 2 boundarypoint remains mysterious. Recent examples of M. Saito [S6] and the secondauthor [Pe3] show that there are analytic obstructions which prevent ANFs fromsurjecting onto (or even mapping nontrivially to) the putative singularity groupfor ANFs (rational.0; 0/ classes in the local intersection cohomology). At firstglance this appears to throw the existence of singularitiesfor Griffiths–Greennormal functions (and hence the Hodge conjecture) into serious doubt, but in~ 5.5 we show that this concern is probably ill-founded.

The last section is devoted to a discussion of Mumford–Tate groups of mixedHodge structures (introduced by Y. Andre [An]) and variations thereof, in partic-ular those attached to admissible normal functions. The motivation for writingthis section was again to attempt to “force singularities toexist” via conditionson the normal function (e.g., involving the zero locus) which maximize the mon-odromy of the underlying local system inside the M-T group; we were able tomarkedly improve Andre’s maximality result (but not to produce singularities).Since the general notion of (non)singularity of a VMHS at a boundary point isdefined here (in~ 6.3), which generalizes the notion of singularity of a normalfunction, we should point out that there is another sense in which the word“singularity” is used in this paper. The “singularities” ofa period mappingassociated to a VHS or VMHSare points where the connection has poles orthe local system has monodromy (˙ in the notation above), and at which onemust compute a limit mixed Hodge structure (LMHS). These contain the “sin-gularities of the VMHS”, nearly always as aproper subset; indeed, pure VHSnever have singularities (in the sense of~ 6.3), though their corresponding periodmappings do.

This paper has its roots in the first author’s talk at a conference in honor ofPhillip Griffiths’ 70th birthday at the IAS, and the second author’s talk at MSRIduring the conference on the topology of stratified spaces towhich this volumeis dedicated. The relationship between normal functions and stratifications oc-curs in the context of mixed Hodge modules and the Decomposition Theorem


[BBD], and is most explicitly on display in the constructionof the multivariableNeron model in~ 5 as a topological group whose restrictions to the strata of aWhitney stratification are complex Lie groups. We want to thank the conferenceorganizers and Robert Bryant for doing an excellent job at putting together andhosting a successful interdisciplinary meeting blending (amongst other topics)singularities and topology of complex varieties,L2 and intersection cohomol-ogy, and mixed Hodge theory, all of which play a role below. Weare indebtedto Patrick Brosnan, Phillip Griffiths, and James Lewis for helpful conversationsand sharing their ideas. We also want to thank heartily both referees as well asChris Peters, whose comments and suggestions have made thisa better paper.

One observation on notation is in order, mainly for experts:to clarify the dis-tinction in some places between monodromy weight filtrations arising in LMHSand weight filtrations postulated as part of the data of an admissible variationof mixed Hodge structure (AVMHS), the former are always denoted M� (andthe latterW�) in this paper. In particular, for a degeneration of (pure) weightnHS with monodromy logarithmN , the weight filtration on the LMHS is writtenM.N /� (and centered atn). While perhaps nontraditional, this is consistent withthe notationM.N;W /� for relative weight monodromy filtrations for (admissi-ble) degenerations of MHS. That is, whenW is “trivial” ( WnDH, Wn�1Df0g)it is simply omitted.

Finally, we would like to draw attention to the interesting recent article [Gr4]of Griffiths which covers ground related to our~ ~ 2–5, but in a complementaryfashion that may also be useful to the reader.

1. Prehistory and classical results

The present chapter is not meant to be heroic, but merely aimsto introducea few concepts which shall be used throughout the paper. We felt it wouldbe convenient (whatever one’s background) to have an up-to-date, “algebraic”summary of certain basic material on normal functions and their invariants inone place. For background or further (and better, but much lengthier) discussionof this material the reader may consult the excellent books [Le1] by Lewis and[Vo2] by Voisin, as well as the lectures of Green and Voisin from the “Torinovolume” [GMV] and the papers [Gr1; Gr2; Gr3] of Griffiths.

Even experts may want to glance this section over since we have includedsome bits of recent provenance: the relationship between log-infinitesimal andtopological invariants, which uses work of M. Saito; the result on inhomoge-neous Picard–Fuchs equations, which incorporates a theorem of Muller-Stachand del Angel; the important example of Morrison and Walcherrelated to openmirror symmetry; and the material onK-motivation of normal functions (see~ 1.3 and~ 1.7), which will be used in Sections 2 and 4.


Before we begin, a word on thecurrentsthat play a role in the bullet-trainproof of Abel’s Theorem in~ 1.1. These are differential forms with distributioncoefficients, and may be integrated againstC 1 forms, with exterior deriva-tive d defined by “integration by parts”. They form a complex computing C-cohomology (of the complex manifold on which they lie) and includeC 1chainsand log-smooth forms. For example, for aC 1 chain� , the delta currentı�has the defining property

R

ı� ^ ! DR

� ! for any C 1 form !. (For moredetails, see Chapter 3 of [GH].)

1.1. Abel’s Theorem. Our (historically incorrect) story begins with a divisorD of degree zero on a smooth projective algebraic curveX=C; the associatedanalytic varietyX an is a Riemann surface. (Except when explicitly mentioned,we continue to work overC.) Writing D D

P

finite nipi 2 Z1.X /hom (ni 2 Z

such thatP

ni D 0, pi 2 X.C/), by Riemann’s existence theorem one has ameromorphic1-form O! with Respi

. O!/D ni (8i). Denoting byf!1; : : : ; !gg abasis for 1.X /, consider the map

Z1.X /hom//

eAJ**

˝1.X /_R

H1.X ;Z/. � /

evf!i g

Š

//

Cg

�2gDW J 1.X /

Dffl

//

R

�ffl

//

�R

� !1; : : : ;R

� !g

�

(1-1)

where� 2C1.Xan/ is any chain with@� DD andJ 1.X / is theJacobianof X .

The1-current� WD O!� 2� iı� is closed; moreover, ifeAJ .D/D 0 then� maybe chosen so that all

R

� !i D 0 impliesR

X �^!i D 0. We can therefore smooth� in its cohomology class to!D ��d� (! 2˝1.X /; �2D0.X /D0-currents),and

f WD exp˚R

. O! �!/

(1-2)

D e2� iR

ı� e� (1-3)

+3

−1 −2

D

Γ


is single-valued — though possibly discontinuous — by (1-3), while being mero-morphic — though possibly multivalued — by (1-2). Locally atpi , e

R

.ni=z/dzD

C zni has the right degree; and so the divisor off is preciselyD. Conversely,if D D .f /D f �1.0/�f �1.1/ for f 2 C.X /�, then

t ‘

Z

f �1.�!0:t /

. � /

induces a holomorphic mapP1! J 1.X /. Such a map is necessarily constant(say, to avoid pulling back a nontrivial holomorphic1-form), and by evaluatingat t D 0 one finds that this constant is zero. So we have proved part (i)of

THEOREM 2. (i) [Abel] Writing Z1.X /rat for the divisors of functionsf 2C.X /�, eAJ descends to an injective homomorphism of abelian groups

CH 1.X /hom WDZ1.X /hom

Z1.X /rat

AJ� J 1.X /:

(ii) [Jacobi inversion]AJ is surjective; in particular, fixing q1; : : : ; qg 2 X.C/

the morphismSymg X ! J 1.X / induced byp1C� � �Cpg‘R

@�1.P

pi �qi /. � /

is birational.

Here@�1D means any1-chain bounding onD. Implicit in (ii) is that J 1.X / isan (abelian) algebraic variety; this is a consequence of ampleness of the thetaline bundle (onJ 1.X /) induced by the polarization

Q W H 1.X;Z/�H 1.X;Z/! Z

(with obvious extensions toQ, R, C) defined equivalently by cup product, inter-section of cycles, or integration.!; �/‘

R

X ! ^ �. The ampleness boils downto the second Riemann bilinear relation, which says thatiQ. � ; N� / is positivedefinite on˝1.X /.

1.2. Normal functions. We now wish to vary the Abel–Jacobi map in families.Until ~ 2, all our normal functions shall be over a curveS . Let X be a smoothprojective surface, andN� W X ! S a (projective) morphism which is

(a) smooth off a finite set D fs1; : : : ; seg � S , and(b) locally of the form.x1;x2/‘ x1x2 at singularities (ofN�).Write Xs WD N�

�1.s/ (s 2 S) for the fibers. The singular fibersXsi(i D

1; : : : ; e) then have only nodal (ordinary double point) singularities, and writingX � for their complement we have� W X � ! S� WD Sn˙ . Fixing a generals0 2 S�, the local monodromiesTsi

2 Aut�

H 1.Xs0;Z/DWHZ;s0

�

of the localsystemHZ WDR1��ZX� are then computed by the Picard–Lefschetz formula

.Tsi� I/ D

X

j

. � ıj /ıj : (1-4)


π

Herefıj g are the Poincare duals of the (possibly nondistinct) vanishing cycleclasses2 ker

˚

H1.Xs0;Z/!H1.Xsi

;Z/

associated to each node onXsi; we

note .Tsi� I/2 D 0. For a family of elliptic curves, (1-4) is just the familiar

Dehn twist:

T

α

β

Τ(α)=α

Τ(β)=β+α β

β+α

∆

Ε Εs 00

0s 0

(For the reader new to such pictures, the two crossing segments in the “localreal” picture at the top of the page become the two touching “thimbles”, i.e., asmall neighborhood of the singularity inE0, in this diagram.)

Now, in our setting, the bundle of JacobiansJ WDS

s2S� J 1.Xs/ is a complex(algebraic) manifold. It admits a partial compactificationto a fiber space ofcomplex abelian Lie groups, by defining

J 1.Xsi/ WD

H 0.!Xsi/

im fH 1.Xsi;Z/g

(with !xsthe dualizing sheaf) andJe WD

S

s2S J 1.Xs/. (How this is topolo-gized will be discussed in a more general context in~ 5.) The same notation will


denote their sheaves of sections,

0!HZ! F_! J ! 0 .on S�/ (1-5)

0!HZ;e! .Fe/_! Je! 0 .on S/; (1-6)

with F WD ��!X=S , Fe WD N��!X=S , HZ DR1��Z, HZ;e DR1 N��Z.

DEFINITION 3. A normal function(NF) is a holomorphic section (overS�) ofJ . An extended(or Poincare) normal function(ENF) is a holomorphic section(overS) of Je. An NF isextendableif it lies in imfH 0.S;Je/!H 0.S�;J /g.

Next consider the long exact cohomology sequence (sectionsoverS�)

0!H 0.HZ/!H 0.F_/!H 0.J /!H 1.HZ/!H 1.F_/I (1-7)

the topological invariantof a normal function� 2 H 0.J / is its imageŒ�� 2H 1.S�;HZ/. It is easy to see that the restriction ofŒ�� to H 1.��

i ;HZ/ (�i apunctured disk aboutsi) computes the local monodromy.Tsi

� I/ Q� (where Q�is a multivalued local lift of� to F_), modulo the monodromy of topologicalcycles. We say that� is locally liftable if all these restrictions vanish, i.e.,if

.Tsi� I/ Q� 2 im f.Tsi

� I/HZ;s0g:

Together with the assumption that as a (multivalued, singular) “section” ofF_e ,

Q�e has at worst logarithmic divergence atsi (the “logarithmic growth” in thetitle), this is equivalent to extendability.

1.3. Normal functions of geometric origin. Let Z 2 Z1.X /prim be a divisorproperly intersecting fibers ofN� and avoiding its singularities, and which isprimitive in the sense that eachZs WD Z �Xs (s 2 S�) is of degree 0. (In fact,the intersection conditions can be done away with, by movingthe divisor in arational equivalence.) Thens‘ AJ.Zs/ defines a section�Z of J , and it canbe shown that a multipleN�ZD �N Z of �Z is always extendable. One says that�Z itself is admissible.

Now assumeN� has a section� W S ! X (also avoiding singularities) andconsider the analog of (1-7) forJe

0!H 0.F_

e /

H 0.HZ;e/!H 0.Je/! ker

˚

H 1.HZ;e/!H 1.F_

e /

! 0:

With a bit of work, this becomes

0! J 1.X=S/fix// ENF

Œ � �//

Hg1.X /prim

Z˝

ŒXs0�˛ ! 0; (1-8)

where the Jacobian of the fixed partJ 1.X=S/fix Œ J 1.Xs/ (8s 2 S) gives aconstant subbundle ofJe and the primitive Hodge classes Hg1.X /prim are the


Q-orthogonal complement of a general fiberXs0of N� in Hg1.X / WDH 2.X ;Z/\

H 1;1.X ;C/.

PROPOSITION4. Let� be an ENF.

(i) If Œ��D 0 then� is a constant section ofJfix WDS

s2S J 1.X=S/fix � Je.(ii) If .� D/�Z is of geometric origin, thenŒ�Z�D ŒZ� (ŒZ�D fundamental class).(iii) [Poincare Existence Theorem]Every ENF is of geometric origin.

We note that (i) follows from considering sectionsf!1; : : : ; !gg.s/ of F_e whose

restrictions to generalXs are linearly independent (such do exist), evaluating alift Q� 2H 0.F_

e / against them, and applying Liouville’s Theorem. The resultingconstancy of the abelian integrals, by a result in Hodge Theory (cf. end of~ 1.6),implies the membership of�.s/2Jfix . To see (iii), apply “Jacobi inversion withparameters” andqi.s/D �.s/ (8i) overS� (really, over the generic point ofS),and then take Zariski closure.1 Finally, when� is geometric, the monodromiesof a lift Q� (to F_

e ) around each loop inS (which determineŒ��) are just the cor-responding monodromies of a bounding1-chain�s (@�s DZs), which identifywith the Leray.1; 1/ component ofŒZ� in H 2.X /; this gives the gist of (ii).

T∆ 0s 0

A normal function is said to bemotivated overK (K � C a subfield) if it isof geometric origin as above, and if the coefficients of the defining equations ofZ, X , N�, andS belong toK.

1.4. Lefschetz (1,1) Theorem.Now takeX � PN to be a smooth projectivesurface of degreed , andfXs WD X �Hsgs2P1 a Lefschetz pencilof hyperplanesections: the singular fibers have exactly one (nodal) singularity. Letˇ WX “ X

denote the blow-up at the base locusB WDT

s2P1 Xs of the pencil, andN� WX ! P1 DW S the resulting fibration. We are now in the situation consideredabove, with�.S/ replaced byd sectionsE1q� � �qEd D ˇ

�1.B/, and fibersof genusg D

�

d�12

�

; and with the added bonus that there is no torsion in any

1Here theqi .s/ are as in Theorem 2(ii) (but varying with respect to a parameter). If at a generic point�.�/ is a special divisor then additional argument is needed.


H 1.��

i ;HZ/, so that admissible) extendable. Hence, givenZ 2Z1.X /prim

(deg.Z �Xs0/D 0): ˇ�Z is primitive,vZ WD �ˇ�Z is an ENF, andŒvZ �Dˇ

�ŒZ�

underˇ� W Hg1.X /primŒ Hg1.X /prim=ZhŒXs0�i.

If, on the other hand, we start with a Hodge class� 2Hg1.X /prim, ˇ�� is (by(1-8) + Poincare existence) the class of a geometric ENF�Z; andŒZ�� Œ�Z��ˇ

��

mod ZhŒXs0�i implies � � ˇ�ˇ

�� Œˇ�Z DW Z� in Hg1.X /=ZhŒXs0�i, which

implies � D ŒZ0� for someZ0 2 Z1.X /.prim/. This is the gist of Lefschetz’soriginal proof [L] of

THEOREM 5. LetX be a(smooth projective algebraic) surface. The fundamen-

tal class mapCH 1.X /Œ � �! Hg1.X / is (integrally) surjective.

This continues to hold in higher dimension, as can be seen from an inductivetreatment with ENF’s or (more easily) from the “modern” treatment of Theo-rem 5 using the exponential exact sheaf sequence

0! ZX �OXe2�i. � /

� O�

X ! 0:

One simply puts the induced long exact sequence in the form

0!H 1.X;O/

H 1.X;Z/!H 1.X;O�/! ker

˚

H 2.X;Z/!H 2.X;O/

! 0;

and interprets it as

0 // J 1.X / //

n

holomorphicline bundles

o

››

ffl

ffl

ffl

// Hg1.X / // 0

CH 1.X /

99

s

s

s

s

s

s

s

s

s

s

s

(1-9)

where the dotted arrow takes the divisor of a meromorphic section of a givenbundle. Existence of the section is a standard but nontrivial result.

We note that forX ! P1 a Lefschetz pencil ofX , in (1-8) we have

J 1.X=P1/fix D J 1.X / WD

H 1.X;C/

F1H 1.X;C/CH 1.X;Z/;

which is zero ifX is a complete intersection; in that caseENF is finitely gen-erated and � embeds Hg1.X /prim in ENF.

EXAMPLE 6. For X a cubic surface� P3, divisors with support on the27

lines already surject onto Hg1.X / D H 2.X;Z/ Š Z7. Differences of theselines generate all primitive classes, hence all of im.ˇ�/ (ŠZ6) in ENF .ŠZ8).


Note thatJe is essentially an elliptic surface andENF comprises the (holomor-phic) sections passing through theC�’s over points of . There are no torsionsections.

1.5. Griffiths’ AJ map. A Z-Hodge structure (HS) of weightm comprises afinitely generated abelian groupHZ together with a descending filtrationF � onHC WDHZ˝Z C satisfyingFpHC˚Fm�pC1HC DHC, theHodge filtration;we denote the lot byH . Examples include them-th (singular/Betti + de Rham)cohomology groups of smooth projective varieties overC, with FpH m

dR.X;C/

being that part of the de Rham cohomology represented byC 1 forms onX an

with at leastp holomorphic differentials wedged together in each monomialterm. (These are forms ofHodge type.p;m�p/C .pC 1;m�p � 1/C � � � ;note thatH p;m�p

C WDFpHC\Fm�pHC.) To accommodateH m of nonsmoothor incomplete varieties, the notion of a (Z-)mixed Hodge structure (MHS)V isrequired: in addition toF � on VC, introduce a decreasingweight filtrationW�

on VQ such that the�

GrWi VQ; .GrWi .VC;F�//�

areQ-HS of weighti . MixedHodge structures have Hodge group

Hgp.V / WD kerfVZ˚FpW2pVC! VCg

(for for VZ torsion-free becomesVZ\FpW2pVC) and Jacobian group

J p.V / WDW2pVC

FpW2pVCCW2pVQ\VZ

;

with special cases Hgm.X / WDHgm.H 2mX // andJ m.X / WDJ m.H 2m�1.X //.Jacobians of HS yield complex tori, and subtori correspond bijectively to sub-HS.

A polarizationof a Hodge structureH is a morphismQ of HS (defined overZ; complexification respectsF �) from H�H to the trivial HSZ.�m/ of weight2m (and type.m;m/), such that viewed as a pairingQ is nondegenerate andsatisfies a positivity constraint generalizing that in~ 1.1 (thesecond Hodge–Riemann bilinear relation). A consequence of this definition is that underQ,Fp is the annihilator ofFm�pC1 (thefirst Hodge–Riemann bilinear relationinabstract form). IfX is a smooth projective variety of dimensiond , Œ˝� the classof a hyperplane section, write (fork � d , say)

H m.X;Q/prim WD kerfH m.X;Q/[˝d�kC1

� H 2d�mC2.X;Q/g:

This Hodge structure is then polarized byQ.�; �/ WD .�1/.m2 /R

X �^�^˝d�k ,Œ˝� the class of a hyperplane section (obviously since this is aQ-HS, the polar-ization is only defined overQ).


Let X be a smooth projective.2m�1/-fold; we shall consider some equiva-lence relations on algebraic cycles of codimensionm onX . Writing Zm.X / forthe free abelian group on irreducible (complex) codimension p subvarieties ofX , two cyclesZ1;Z2 2Zm.X / are homologically equivalent if their differencebounds aC 1 chain� 2C

top2m�1

.X anIZ/ (of real dimension2m�1). Algebraicequivalence is generated by (the projection toX of) differences of the formW �.X�fp1g/�W �.X�fp2g/whereC is an algebraic curve,W 2Zm.X�C /,andp1;p2 2C.C/ (or C.K/ if we are working over a subfieldK�C). Rationalequivalence is obtained by takingC to be rational (C Š P1), and formD 1 isgenerated by divisors of meromorphic functions. We writeZm.X /rat for cycles�rat 0, etc. Note that

CH m.X / WDZm.X /

Zm.X /rat� CH m.X /hom WD

Zm.X /hom

Zm.X /rat

and

CH m.X /hom� CH m.X /alg WDZm.X /alg

Zm.X /rat

are proper inclusions in general.Now letW �X �C be an irreducible subvariety of codimensionm, with �X

and�C the projections from a desingularization ofW to X andC . If we putZi WD �X�

��

Cfpig, thenZ1 �alg Z2 implies Z1 �hom Z2, which can be seen

explicitly by setting� WD �X��

C.�!q:p/ (so thatZ1�Z2 D @� ).

qp

W

X

C

+

+

−

−

Zπ

πC

X

Let! be ad-closed form of Hodge type.j ; 2m�j �1/ on X , for j at leastm. Consider

R

� ! DR qp �, where� WD �C�

��

X! is ad-closed1-current of type

.j �mC 1;m� j / as integration along the.m� 1/-dimensional fibers of�C

eats up.m� 1;m� 1/. So� D 0 unlessj D m, and by a standard regularitytheorem in that case� is holomorphic. In particular, ifC is rational, we haveR

� ! D 0. This is essentially the reasoning behind the following result:


PROPOSITION7. The Abel–Jacobi map

CH m.X /homAJ

//

�

FmH 2m�1.X;C/�_

R

H2m�1.X ;Z/. � /

Š J m.X / (1-10)

induced byZ D @� ‘R

� . � /, is well-defined and restricts to

CH m.X /algAJalg

//

FmH 2m�1hdg .X;C/

R

H2m�1.X ;Z/. � /

Š J m.H 2m�1hdg .X //DW J m

h.X /; (1-11)

whereH 2m�1hdg .X / is the largest sub-HS ofH 2m�1.X / contained(after tensor-

ing with C) in H m�1;m.X;C/˚H m;m�1.X;C/. While J m.X / is in generalonly a complex torus(with respect to the complex structure of Griffiths), J m

h.X /

is an abelian variety. Further, assuming a special case of the generalized Hodgeconjecture, if X is defined overk thenJ m

h.X / andAJalg.Z/ are defined overNk.

REMARK 8. (i) To see thatJ mh.X / is an abelian variety, one uses the Kodaira

embedding theorem: by the Hodge–Riemann bilinear relations, the polarizationof H 2m�1.X / induces a Kahler metrich.u; v/ D �iQ.u; Nv/ on J m

h.X / with

rational Kahler class.

(ii) The mapping (1-10) is neither surjective nor injectivein general, and (1-11)is not injective in general; however, (1-11) is conjecturedto be surjective, andregardless of thisJ m

alg.X / WD im.AJalg/�J mh.X / is in fact a subabelian variety.

(iii) A point in J m.X / is naturally the invariant of an extension of MHS

0! .H D/H 2m�1.X;Z.m//!E! Z.0/! 0

(where the “twist” Z.m/ reduces weight by2m, to .�1/). The invariant isevaluated by taking two lifts�F 2 F0W0EC, �Z 2W0EZ of 1 2 Z.0/, so that�F � �Z 2 W0HC is well-defined modulo the span ofF0W0HC and W0HZ

hence is inJ 0.H /Š J m.X /. The resulting isomorphism

J m.X /Š Ext1MHS.Z.0/;H2m�1.X;Z.m///

is part of an extension-class approach toAJ maps (and their generalizations)due to Carlson [Ca].

(iv) The Abel–Jacobi map appears in [Gr3].

1.6. Horizontality. Generalizing the setting of~ 1.2, letX be a smooth projec-tive 2m-fold fibered over a curveS with singular fibersfXsi

g each of either

(i) NCD (normal crossing divisor) type: locally.x1; : : : ;x2m/�‘Qk

jD1 xj ; or


(ii) ODP (ordinary double point) type: locally.x/‘P2m

jD1 x2j .

An immediate consequence is that allTsi2Aut

�

H 2m�1.Xs0;Z/

�

areunipo-tent: .Tsi

�I/nD 0 for n� 2m in case (i) orn� 2 in case (ii). (If all fibers areof NCD type, then we say the familyfXsg of .2m� 1/-folds issemistable.)

The Jacobian bundle of interest isJ WDS

s2S� J m.Xs/ .� Jalg/. Writing˚

F.m/ WD R

2m�1��˝��mX�=S�

�˚

H WD R2m�1��˝

�

X�=S�

�˚

HZ WDR2m�1��ZX�

;

and notingF_ŠH=F viaQ W H2m�1�H2m�1!OS� , the sequences (1-5) and(1-7), as well as the definitions of NF and topological invariantŒ � �, all carry over.A normal function of geometric origin, likewise, comes fromZ 2 Zm.X /prim

with Zs0WD Z �Xs0

�hom 0 (on Xs0), but now has an additional feature known

ashorizontality, which we now explain.Working locally over an analytic ballU � S� containings0, let

Q! 2 � .XU ;FmC1˝2m�1

X1 /

be a “lift” of !.s/ 2 � .U;FmC1/, and�s 2 Ctop2m�1

.XsIZ/ be a continuousfamily of chains with@�s D Zs. Let P " be a path froms0 to s0 C "; thenO� " WD

S

s2P " �s has boundary�s0C"��s0CS

s2P " Zs, and�

@

@s

Z

�s

!.s/

�

sDs0

D lim"!0

1

"

Z

�s0C"��s0

Q!

D lim"!0

1

"

�Z

@ O� "

Q! �

Z s0C"

s0

Z

Zs

!.s/

�

D

Z

�s0

˝

Ad=dt ; d Q!˛

�

Z

Zs0

!.s0/; (1-12)

where��Ad=dt D d=dt (with Ad=dt tangent to O� ", OZ").

The Gauss–Manin connectionr W H!H˝˝1S� differentiates the periods of

cohomology classes (against topological cycles) in families, satisfies Griffithstransversalityr.Fm/� Fm�1˝˝1

S� , and is computed by

r! D�˝

Ad=dt ; d Q!˛�

˝ dt:

Moreover, the pullback of any form of typeFm to Zs0(which is of dimension

m � 1) is zero, so thatR

Zs0

!.s0/ D 0 andR

�s0

r! is well-defined. If Q� 2

� .U;H/ is any lift of AJ.�s/ 2 � .U;J /, we therefore have

Q�

rd=dtQ� ; !

�

Dd

dsQ. Q� ; !/�Q. Q� ;r!/D

d

ds

Z

�s

! �

Z

�s

r!;


which is zero by (1-12) and the remarks just made. We have shown thatrd=dtQ�

kills FmC1, and sord=dtQ� is a local section ofFm�1.

DEFINITION 9. A normal function� 2 H 0.S�;J / is horizontal if r Q� 2� .U;Fm�1 ˝ ˝1

U/ for any local lift Q� 2 � .U;H/. Equivalently, if we set

Hhor WD ker�

Hr

! H

Fm�1 ˝˝1S�

�

� Fm DW F , .F_/hor WDHhorF

, andJhor WD.F_/hor

HZ, then NFhor WDH 0.S;Jhor/.

Much as anAJ image was encoded in a MHS in Remark 8(ii), we may encodehorizontal normal functions in terms of variations of MHS. AVMHS V=S�

consists of aZ-local systemV with an increasing filtration ofVQ WD VZ˝Z Q

by sublocal systemsWiVQ, a decreasing filtration ofV.O/ WD VQ˝Q OS� byholomorphic vector bundlesFj .DFjV/, and a connectionr W V! V˝˝1

S�

such thatr.V/ D 0, the fibers.Vs;W�;Vs;F�

s / yield Z-MHS, andr.Fj / �

Fj�1 ˝ ˝1S� (transversality). (Of course, a VHS is just a VMHS with one

nontrivial GrWi VQ, and..HZ;H;F�/;r/ in the geometric setting above gives

one.) A horizontal normal function corresponds to an extension

0!

wt:�1VHS‚…„ƒ

H.m/! E ! Z.0/S� ! 0 (1-13)

“varying” the setup of Remark 8(iii), with the transversality of the lift of �F .s/

(together with flatness of�Z.s/) reflecting horizontality.

REMARK 10. Allowing the left-hand term of (1-13) to have weight lessthan�1

yields “higher” normal functions related to families ofgeneralized(“higher”)algebraic cycles. These have been studied in [DM1; DM2; DK],and will beconsidered in later sections.

An important result on VHS over a smooth quasiprojective base is that the globalsectionsH 0.S�;V/ (resp.H 0.S�;VR/, H 0.S�;VC/) span theQ-local system(resp. its tensor product withR, C) of a (necessarily constant) sub-VMHS� V ,called thefixed partVfix (with constant Jacobian bundleJfix).

1.7. Infinitesimal invariant. Given � 2 NFhor, the “r Q�” for various localliftings patch together after going modulorFm � Fm�1˝˝1

S� . If r Q� Drffor f 2 � .U;Fm/, then the alternate liftQ� � f is flat, i.e., equals

P

i ci i

wheref ig � � .U;VZ/ is a basis and theci are complex constants. Since the

composition (s 2 S�) H 2m�1.Xs;R/ŒH 2m�1.Xs;C/“H 2m�1.Xs;C/

F m is anisomorphism, we may take theci 2 R, and then they are unique inR=Z. Thisimplies thatŒ�� lies in the torsion group ker

�

H 1.HZ/!H 1.HR/�

, so that amultiple N� lifts to H 0.S�;HR/ � Hfix . This motivates the definition of an


infinitesimal invariant

ı� 2H1�

S�;Fm r

! Fm�1˝˝1S�

� if S�

affineH 0

�

S; Fm�1˝˝1

Fm

�

(1-14)

as the image of� 2H 0�

S�; HhorF

�

under the connecting homomorphism inducedby

0! Cone�

Fm r

! Fm�1˝˝1�

Œ�1�! Cone�

Hr

!H˝˝1�

Œ�1�

!Hhor

F! 0: (1-15)

PROPOSITION11. If ı� D 0, then up to torsion, Œ�� D 0 and� is a (constant)section ofJfix .

An interesting application to the differential equations satisfied by normal func-tions is essentially due to Manin [Ma]. For simplicity letS D P1, and supposeH is generated by! 2 H 0.S�;F2m�1/ as aD-module, with monicPicard–Fuchs operatorF.rıs WDs d

ds/ 2 C.P1/�Œrıs

� killing !. Then its periods satisfy

the homogeneous P-F equationF.ıs/R

i! D 0, and one can look at the multi-

valued holomorphic functionQ. Q�; !/ (whereQ is the polarization, andQ� is amultivalued lift of � to Hhor=F), which in the geometric case is just

R

�s!.s/.

The resulting equation

.2� i/mF.ıs/Q. Q�; !/DWG.s/ (1-16)

is called theinhomogeneous Picard–Fuchs equationof �.

PROPOSITION12. (i) [DM1] G 2C.P1/� is a rational function holomorphic onS�; in theK-motivated setting(taking also! 2 H 0.P1; N��!X=P1/, and hence

F , overK), G 2K.P1/�.

(ii) [Ma; Gr1] G � 0 ” ı� D 0.

EXAMPLE 13. [MW] The solutions to

.2� i/2n

ı4z � 5z

4Q

`D1

.5ızC `/o

. � /D�15

4

pz

are the membrane integralsR

�s!.s/ for a family of 1-cycles on the mirror

quintic family of Calabi–Yau3-folds. (The family of cycles is actually onlywell-defined on the double-cover of this family, as reflectedby the

pz.) What

makes this example particularly interesting is the “mirrordual” interpretationof the solutions as generating functions of open Gromov–Witten invariants of afixed Fermat quintic3-fold.


The horizontality relationr Q� 2 Fm�1 ˝ ˝1 is itself a differential equation,and the constraints it puts on� over higher-dimensional bases will be studied in~ 5.4–5.

Returning to the setting described in~ 1.6, there arecanonical extensionsHe

andF�

e of H;F� across thesi as holomorphic vector bundles or subbundles(reviewed in~ 2 below); for example, if all fibers are of NCD type thenFp

e Š

R2m�1 N��˝��p

X=S.log.XnX �//. Writing2

HZ;e WDR2m�1 N��ZX and He;hor WD ker

�

Her

!He

Fm�1e

˝˝1S .log˙/

�

;

we have short exact sequences

0!HZ;e!He.;hor/

Fme

! Je.;hor/! 0 (1-17)

and setENF.hor/ WDH 0.S;Je.;hor//.

THEOREM 14. (i) Z 2Zm.X /prim impliesN�Z 2 ENFhor for someN 2 N.(ii) � 2 ENFhor with Œ�� torsion impliesı� D 0.

REMARK 15. (ii) is essentially a consequence of the proof of Corollary 2 in[S2]. For� 2 ENFhor, ı� lies in the subspace

H1�

S;Fm r

! Fm�1e ˝˝1

S .log˙/�

;

the restriction of

H1�

S�;Fm r

! Fm�1˝˝1

S�

�

!H 1.S�;HC/

to which is injective.

1.8. The Hodge Conjecture? Putting together Theorem 14(ii) and Proposi-tion 12, we see that a horizontal ENF with trivial topological invariant lies inH 0.S;Jfix/DWJ

m.X=S/fix (constant sections). In fact, the long exact sequenceassociated to (17) yields

0! J m.X=S/fix! ENFhorŒ � �!

Hgm.X /prim

im˚

Hgm�1.Xs0/ ! 0;

with Œ�Z� D ŒZ� (if �Z 2 ENF) as before. IfXN�! P1 D S is a Lefschetz pencil

on a2m-fold X , this becomes

2Warning: whileHe has no jumps in rank, the stalk ofHZ;e at si 2˙ is of strictly smaller rank than ats 2 S�.


J m.X /ffl

�

// ENFhorŒ � �

.�/// // Hgm.X /prim˚ ker

n

Hgm�1.B/!Hgm.X /

o

CH m.X /prim

�. � /

OO

ker.Œ � �/

AJ

OO

ffl

�

// CH m.X /prim

v. � /

cc

Œ � �

.��///

ˇ�

OO

Hgm.X /prim?

ffl

.id;0/

OO

(1-18)

where the surjectivity of.�/ is due to Zucker (compare Theorems 31 and 32 in~ 3 below; his result followed on work of Griffiths and Bloch establishing thesurjectivity forsufficiently ampleLefschetz pencils). What we are after (modulotensoring withQ) is surjectivity of the fundamental class map.��/. This wouldclearly follow from surjectivity of�. � /, i.e., a Poincare existence theorem, as in~ 1.4. By Remark 8(ii) this cannot work in most cases; however we have this:

THEOREM 16. The Hodge Conjecture HC.m;m/ is true for X if J m.Xs0/ D

J m.Xs0/alg for a general member of the pencil.

EXAMPLE 17 [Zu1]. AsJ 2 D J 2alg is true for cubic threefolds by the work of

Griffiths and Clemens [GC], HC.2; 2/ holds for cubic fourfolds inP5.

The Lefschetz paradigm, of taking a1-parameter family of slices of a primi-tive Hodge class to get a normal function and constructing a cycle by Jacobiinversion, appears to have led us (for the most part) to a deadend in highercodimension. A beautiful new idea of Griffiths and Green, to be described in~ 3, replaces the Lefschetz pencil by a complete linear system(of higher degreesections ofX ) so that dim.S/� 1, and proposes to recover algebraic cyclesdual to the given Hodge class from features of the (admissible) normal functionin codimension� 2 on S .

1.9. Deligne cycle-class.This replaces the fundamental andAJ classes byone object. WritingZ.m/ WD .2� i/mZ, define the Deligne cohomology ofX

(smooth projective of any dimension) by

H �

D.Xan;Z.m// WD

H ��

Cone˚

C �

top.XanIZ.m//˚Fm

D�.X an/!D�.X an/

Œ�1��

;

andcD WCH m.X /!H 2mD.X;Z.m// by Z‘ .2� i/m.Ztop; ıZ ; 0/. One easily

derives the exact sequence

0! J m.X /!H 2mD .X;Z.m//! Hgm.X /! 0;

which invites comparison to the top row of (1-18).


2. Limits and singularities of normal functions

Focusing on the geometric case, we now wish to give the readera basic intu-ition for many of the objects — singularities, Neron models, limits of NF’s andVHS — which will be treated from a more formal Hodge-theoretic perspectivein later sections.3 The first part of this section (~ ~ 2.2–8) considers a coho-mologically trivial cycle on a 1-parameter semistably degenerating family ofodd-dimensional smooth projective varieties. Such a family has two invariants“at” the central singular fiber:

� the limit of the Abel–Jacobi images of the intersections of the cycle with thesmooth fibers, and

� the Abel–Jacobi image of the intersection of the cycle with the singular fiber.

We define what these mean and explain the precise sense in which they agree,which involves limit mixed Hodge structures and the Clemens–Schmid exactsequence, and links limits ofAJ maps to the Bloch–Beilinson regulator onhigherK-theory.

In the second part, we consider what happens if the cycle is only assumed tobe homologically trivialfiberwise. In this case, just as the fundamental class ofa cycle on a variety must be zero to define itsAJ class, the family of cycles hasa singularity class which must be zero in order to define the limit AJ invari-ant. Singularities are first introduced for normal functions arising from familiesof cycles, and then in the abstract setting of admissible normal functions (andhigher normal functions). At the end we say a few words about the relationof singularities to the Hodge conjecture, their role in multivariable Neron mod-els, and the analytic obstructions to singularities discovered by M. Saito, topicswhich ~ 3, ~ 5.1–2, and~ 5.3–5, respectively, will elaborate extensively upon.

We shall begin by recastingcD from ~ 1.9 in a more formal vein, which works˝Q. The reader should note that henceforth in this paper, we have to introduceappropriate Hodge twists (largely suppressed in~ 1) into VHS, Jacobians, andrelated objects.

2.1. AJ map. As we saw earlier (Section 1), theAJ map is the basic Hodge-theoretic invariant attached to a cohomologically trivialalgebraic cycle on asmooth projective algebraic varietyX=C; say dim.X /D 2m�1. In the diagramthat follows, ifclX ;Q.Z/D 0 thenZD @� for � (say) a rationalC 1 .2m�1/-chain onX an, and

R

� 2 .FmH 2m�1.X;C//_ inducesAJX ;Q.Z/.

3Owing to our desire to limit preliminaries and/or notational complications here, there are a few unavoid-able inconsistencies of notation between this and later sections.


HomMHS

�

Q.0/;H 2m.X;Q.m//�

.H 2m.X //.m;m/Q

CH m.X /

clX

55

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

// Ext1DbMHS

�

Q.0/;K�Œ2m�.m/�

OO

ker.clX /?

ffl

OO

AJX// Ext1

MHS

�

Q.0/;H 2m�1.X;Q.m//�

OO

J m.X /Q Š

�

FmH 2m�1C

�_

H 2m�1Q.m/

(2-1)

The middle term in the vertical short-exact sequence is isomorphic to Delignecohomology and Beilinson’s absolute Hodge cohomologyH 2m

H.X an;Q.m//,

and can be regarded as the ultimate strange fruit of Carlson’s work on extensionsof mixed Hodge structures. HereK� is a canonical complex of MHS quasi-isomorphic (noncanonically) to

L

i H i.X /Œ�i �, constructed from two generalconfigurations of hyperplane sectionsfHig

2m�1iD0

, f QHj g2m�1jD0

of X . More pre-cisely, looking (forjI j; jJ j > 0) at the corresponding “cellular” cohomologygroups

CI;J

H ; QH.X / WDH 2m�1

�

XnS

i2I

Hi ;S

j2J

Hjn � � � IQ�

;

one setsK` WD

M

I;JjI j�jJ jD`�2mC1

CI;J

H ; QH.X /I

refer to [RS]. (Ignoring the description ofJ m.X / andAJ , and the comparisonsto cD;HD, all of this works for smooth quasiprojectiveX as well; the verticalshort-exact sequence is true even without smoothness.)

The reason for writingAJ in this way is to make plain the analogy to (2-9)below. We now pass back toZ-coefficients.

2.2. AJ in degenerating families. To let AJX .Z/ vary with respect to a pa-rameter, consider a semistable degeneration (SSD) over an analytic disk

X � ffl

�

//

�

››

X

N�

››

X0

››

?

_

{0

oo

S

i Yi

�� ffl

� |// � f0g?

_

oo

(2-2)

where X0 is a reduced NCD with smooth irreducible componentsYi , X issmooth of dimension2m, N� is proper and holomorphic, and� is smooth. Analgebraic cycleZ 2Zm.X / properly intersecting fibers gives rise to a family

Zs WD Z �Xs 2Zm.Xs/ ; s 2�:


Assume0D ŒZ� 2H 2m.X / [which implies0D ŒZs � 2H 2m.Xs/]; then is therea sense in which

lims!0

AJXs.Zs/DAJX0

.Z0/? (2-3)

(Of course, we have yet to say what either side means.)

2.3. Classical example.Consider a degeneration of elliptic curvesEs whichpinches 3 loops in the same homology class to points, yielding for E0 threeP1’sjoined at0 and1 (called a “Neron3-gon” or “Kodaira typeI3” singular fiber).

pinch loops

to points

z

z

z

2

1

3

E Es 0 coordinates:

Denote the total space byEN�!�. One has a family of holomorphic1-forms

!s 2 ˝1.Es/ limiting to fdlog.zj /g

3jD1

on E0; this can be thought of as a

holomorphic section ofR0 N��˝1E=�

.logE0/.There are two distinct possibilities for limiting behaviorwhenZs D ps � qs

is a difference of points. (These do not include the case where one or both ofp0, q0 lies in the intersection of two of theP1’s, since in that caseZ is notconsidered to properly intersectX0.)

Case (I):

psp0

q0

sq

Herep0 andq0 lie in the sameP1 (thej D 1 component, say): in which case

AJEs.Zs/D

Z ps

qs

!s 2 C=Z˝ R

˛s!s;

R

ˇs!s

˛

limits toZ q0

p0

dlog.z1/D logz1.p0/

z1.q0/2 C=2� iZ:


Case (II):

βαps

sq

p

q0

0

??In this case,p0 andq0 lie in different P1 components, in which case0 ¤

ŒZ0�2H 2.X0/ [which impliesŒZ�¤0] and we say thatAJ.Z0/ is “obstructed”.

2.4. Meaning of the LHS of (2-3). If we assume only that0D ŒZ��2H 2m.X �/,then

AJXs.Zs/ 2 J m.Xs/ (2-4)

is defined for eachs 2 ��. We can make this into a horizontal, holomorphicsection of a bundle of intermediate Jacobians, which is whatwe shall meanhenceforth by anormal function(on�� in this case).

Recall the ingredients of a variation of Hodge structure (VHS) over��:

HD ..H;HO;F�/;r/; rFp � F

p�1˝˝1S ; 0!H!

HO

Fm! J ! 0;

whereHDR2m�1��Z.m/ is a local system,HO DH˝Z O�� is [the sheaf ofsections of] a holomorphic vector bundle with holomorphic subbundlesF�, andthese yield HS’sHs fiberwise (notation:Hs D .Hs;Hs.;C/;F

�

s /). Henceforthwe shall abbreviateHO to H.

Then (2-4) yields a section of the intermediate Jacobian bundle

�Z 2 � .��;J /:

Any holomorphic vector bundle over�� is trivial, each trivialization inducingan extension to�. The extensions we want are the “canonical” or “privileged”ones (denoted. � /e); as in~ 1.7, we define an extended Jacobian bundleJe by

0! |�H!He

Fme

! Je! 0: (2-5)

THEOREM 18 [EZ]. There exists a holomorphicN�Z 2 � .�;Je/ extending�Z.

Define lims!0 AJXS.Zs/ WD N�Z.0/ in .Je/0, the fiber over0 of the Jacobian

bundle. To be precise: sinceH 1.�; |�H/D f0g, we can lift theN�Z to a sectionof the middle term of (2-5), i.e., of a vector bundle, evaluate at0, then quotientby .|�H/0.


2.5. Meaning of the RHS of (2-3). Higher Chow groups

CH p.X; n/ WD

�

“admissible, closed” codimension palgebraic cycles onX �An

�

“higher” rational equivalence

were introduced by Bloch to compute algebraicKn-groups ofX , and come with“regulator maps” regp;n to generalized intermediate Jacobians

J p;n.X / WDH 2p�n�1.X;C/

FpH 2p�n�1.X;C/CH 2p�n�1.X;Z.p//:

(Explicit formulas for regp;n have been worked out by the first author with J.Lewis and S. Muller-Stach in [KLM].) The singular fiberX0 has motivic coho-mology groupsH �

M.X0;Z. � // built out of higher Chow groups on the substrata

Y Œ`� WD qjI jD`C1YI WD qjI jD`C1.\

i2I

Yi/;

(which yield a semi-simplicial resolution ofX0). Inclusion induces

{�

0 W CH m.X /hom!H 2mM .X0;Z.m//hom

and we defineZ0 WD {�

0Z. TheAJ map

AJX0W H 2m

M .X0;Z.m//hom! J m.X0/ WDH 2m�1.X0;C/

�

FmH 2m�1.X0;C/C

H 2m�1.X0;Z.m//

�

is built out of regulator maps on substrata, in the sense thatthe semi-simplicialstructure ofX0 induces “weight” filtrationsM� on both sides4 and

GrM�` H 2m

M .X0;Z.m//homGrM

�`AJ

� GrM�` J m.X0/

boils down to

fsubquotient ofCH m.Y Œ`�; `/gregm;`

� fsubquotient ofJ m;`.Y Œ`�/g:

4For the advanced reader, we note that ifM� is Deligne’s weight filtration onH 2m�1.X0;Z.m//, thenM�`J m.X0/ WD Ext1MHS.Z.0/;M�`�1H 2m�1.X0;Z.m///. The definition of theM� filtration onmotivic cohomology is much more involved, and we must refer the reader to [GGK, sec. III.A].


2.6. Meaning of equality in (2-3). Specializing (2-5) to0, we have

. N�Z.0/ 2/ J mlim.Xs/ WD .Je/0 D

.He/0

.Fme /0C .|�H/0

;

where.|�H/0 are the monodromy invariant cycles (and we are thinking of thefiber .He/0 over 0 as the limit MHS ofH, see next subsection). H. Clemens[Cl1] constructed a retraction mapr W X “ X0 inducing

H 2m�1.X0;Z/

�

ıı

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

r�

// H 2m�1.X ;Z/

››

� .��;H/

››

� .�; |�H/

››

.|�H/0�

_

››

H 2m�1lim .Xs;Z/

(2-6)

(where� is a morphism of MHS), which in turn induces

J.�/ W J m.X0/! J mlim.Xs/:

THEOREM 19 [GGK]. lims!0 AJXs.Zs/D J.�/

�

AJX0.Z0/

�

:

2.7. Graphing normal functions. On��, let T W H!H be the counterclock-wise monodromy transformation, which is unipotent since the degeneration issemistable. Hence the monodromy logarithm

N WD log.T /D2m�1X

kD1

.�1/k�1

k.T � I/k

is defined, and we can use it to “untwist” the local system˝Q:

HQ‘ QHQ WD exp

�

�logs

2� iN

�

HQŒHe:

In fact, this yields a basis for, and defines, the privileged extensionHe. More-over, sinceN acts onQHQ, it acts onHe, and therefore on.He/0DH 2m�1

lim .Xs/,inducing a “weight monodromy filtration”M�. Writing HDH 2m�1

lim .Xs;Q.m//,this is the unique filtrationf0g � M�2m � � � � � M2m�2 D H satisfying


N.Mk/ � Mk�2 andN k W GrM�1Ck H

Š

! GrM�1�k H for all k. In general it

is centered about the weight of the original variation (cf. the convention in theIntroduction).

EXAMPLE 20. In the “Dehn twist” example of~ 1.2,N DT �I (with N.˛/D0,N.ˇ/D ˛) so that Q D ˛, Q D ˇ� logs

2� i˛ are monodromy free and yield anO�-

basis ofHe. We haveM�3 D f0g, M�2 DM�1 D h˛i, M0 DH .

REMARK 21. Rationally, ker.N /D ker.T � I/ even whenN ¤ T � I .

By [Cl1], � mapsH 2m�1.X0/ onto ker.N / � H 2m�1lim .Xs/ and is compatible

with the twoM�’s; together with Theorem 19 this implies

THEOREM 22. lims!0 AJXs.Zs/ 2 J m .ker.N // .� J m

lim.Xs//. (Here we re-ally mean ker.T � I/ so thatJ m is defined integrally.)

Two remarks:

� This was not visible classically for curves (J 1.ker.N //D J 1lim.Xs/).

� Replacing.Je/0 by J m.ker.N // yieldsJ 0e, which is a “slit-analytic5 Haus-

dorff topological space” (Je is non-Hausdorff because in the quotient topol-ogy there are nonzero points in.Je/0 that look like limits of points in thezero-section ofJe, hence cannot be separated from0 2 .Je/0.6) This is thecorrect extended Jacobian bundle for graphing “unobstructed” (in the senseof the classical example) or “singularity-free” normal functions. Call this the“pre-Neron-model”.

2.8. Nonclassical example.Take a degeneration of Fermat quintic 3-folds

X D semistable reductionof

�

s4P

jD1

z5j D

4Q

kD0

zk

�

� P4 ��;

so thatX0 is the union of5 P3’s blown up along curves isomorphic toC Dfx5 C y5 C z5 D 0g. Its motivic cohomology groupH 4

M.X0;Q.2//hom has

GrM0 isomorphic to 10 copies of Pic0.C /, GrM�1 isomorphic to 40 copies ofC�,

GrW�2 D f0g, and GrM

�3 ŠKind3.C/. One has a commuting diagram

H 4M.X0;Q.2//hom

AJX0// J 2.X0/Q J 2.ker.N //Q

Kind3.C/

reg2;3

//

?

ffl

OO

C=.2� i/2QIm

//

?

ffl

OO

R

(2-7)

5That is, each point has a neighborhood of the form: open ball about 0 in CaCb intersected with..Canf0g/� Cb/[ .f0g � Cc/, wherec � b.

6See the example before Theorem II.B.9 in [GGK].


and explicit computations with higher Chow precycles in [GGK, ~ 4] lead to theresult:

THEOREM 23. There exists a family of1-cyclesZ 2 CH 2.X /hom;Q such thatZ0 2 M�3H 4

Mand Im.AJX0

.Z0// D D2.p�3/, whereD2 is the Bloch–

Wigner function.

Hence, lims!0 AJXs.Zs/ ¤ 0 and so the generalZs in this family is not

rationally equivalent to zero. The main idea is that the family of cycles limits toa (nontrivial) higher cycle in a substratum of the singular fiber.

2.9. Singularities in 1 parameter. If only ŒZs �D 0 (s 2��), andŒZ��D 0 fails,then

lims!0

AJ is obstructed

and we sayN�Z.s/ has a singularity (ats D 0), measured by the finite group

G ŠIm.TQ� I/\HZ

Im.TZ� I/D

�

Z=3Z in the classical example,.Z=5Z/3 in the nonclassical one.

(The .Z=5Z/3 is generated by differences of lines limiting to distinct compo-nents ofX0.) The Neron model is then obtained by replacingJ.ker.N // (in thepre-Neron-model) by its product withG (this will graphall admissible normalfunctions, as defined below).

The next example demonstrates the “finite-group” (or torsion) nature of singu-larities in the 1-parameter case. In~ 2.10 we will see how this feature disappearswhen there are many parameters.

EXAMPLE 24. Let� 2 C be general and fixed. Then

Cs D fx2Cy2C s.x2y2C �/D 0g

defines a family of elliptic curves (inP1�P1) over�� degenerating to a Neron2-gon ats D 0. The cycle

Zs WD

�

i

r

1C�s

1Cs; 1

�

�

�

�i

r

1C�s

1Cs; 1

�

is nontorsion, with points limiting to distinct components. (See figure on nextpage.)

Hence,AJCs.Zs/DW �.s/ limits to the nonidentity component (ŠC�) of the

Neron model. The presence of the nonidentity component removes the obstruc-tion (observed in case (II) of~ 2.3) to graphing ANFs with singularities.


∆

Ε Εs 00

s 0

Neron

2−gon

α

ΖΖ

s0

T

Τ(ν)=ν+α

ν

Two remarks:

� By tensoring withQ, we can “correct” this: write ; ˇ for a basis forH 1.Cs/

andN for the monodromy log about0, which sends ‘0 andˇ‘2˛. SinceN.�/D˛DN.1

2ˇ/, �� 1

2ˇ will pass through the identity component (which

becomes isomorphic toC=Q.1/ after tensoring withQ, however).� Alternately, to avoid tensoring withQ, one can add a2-torsion cycle like

Ts WD .i�14 ; �

14 /� .�i�

14 ;��

14 /:

2.10. Singularities in 2 parameters.

EXAMPLE 25. Now we will effectively allow� (from the last example) to vary:consider the smooth family

Cs;t WD fx2Cy2C sx2y2C t D 0g

over.��/2. The degenerationst! 0 ands! 0 pinch physically distinct cyclesin the same homology class to zero, so thatC0;0 is anI2; we have obviouslythatN1 DN2 (both send ‘ ˛‘ 0). Take

Zs;t WD

�

i

r

1Ct

1Cs; 1

�

�

�

�i

r

1Ct

1Cs; 1

�

for our family of cycles, which splits between the two components of theI2 at.0; 0/. See figure at top of next page.

Things go much more wrong here. Here are 3 ways to see this:

� try to correct monodromy (as we did in Example 24 with�12ˇ): N1.�/D ˛,

N1.ˇ/D ˛, N2.�/D 0, N2.ˇ/D ˛ implies an impossibility;� in Ts (from Example 1),�1=4 becomes (here).t=s/1=4 — so its obvious ex-

tension isn’t well-defined. In fact, there isno 2-torsion family of cycles withfiber over.0; 0/ a difference of two points in the two distinct components of


T(s,t)

(0,0)(0,t)

(s,0)

ν

2 :ν−>ν

T1:ν−>ν+α

C0;0 (that is, one that limits to have the same cohomology class inH 2.C0;0/

asZ0;0).� take the “motivic limit” of AJ at t D 0: under the uniformization ofCs;0 by

P1 3 z’

�

2z

1� sz2;

2iz

1C sz2

�

;

�

i

s.1Cp

1C s/�

��

i

s.1�p

1C s/�

’Zs;0:

Moreover, the isomorphismC� Š K1.C/ ŠM�1H 2M.Cs;0;Z.1// .3 Zs;0/

sends1Cp

1C s

1�p

1C s2 C

�

to Zs;0, and atsD 0 (considering it as a precycle inZ1.�; 1/) this obviouslyhas a residue.

The upshot is thatnontorsionsingularities appear in codimension 2 and up.

2.11. Admissible normal functions. We now pass to the abstract setting ofa complex analytic manifoldNS (for example a polydisk or smooth projectivevariety) with Zariski open subsetS , writing D D NS n S for the complement.Throughout, we shall assume that�0.S/ is finite and�1.S/ is finitely generated.Let V D .V;V.O/;F�;W�/ be a variation of MHS overS .


Admissibilityis a condition which guarantees (at eachx 2D) a well-definedlimit MHS for V up to the actionF�‘ exp.� logT /F� (� 2C) of local unipo-tent monodromiesT 2 �.�1.Ux \ S//. If D is a divisor with local normalcrossings atx, andV is admissible, then a choice of coordinatess1; : : : ; sm onan analytic neighborhoodU D �k of x (with fs1 � � � sm D 0g D D) producesthe LMHS . sV/x . Here we shall only indicate what admissibility, and thisLMHS, is in two cases: variations of pure HS, and generalizednormal functions(cf. Definition 26).

As a consequence of Schmid’s nilpotent- and SL2-orbit theorems, pure varia-tion is always admissible. IfVDH is a pure variation in one parameter, we have(at least in the unipotent case) already defined “Hlim” and now simply replacethat notation by “. sH/x”. In the multiple parameter (or nonunipotent) setting,simply pull the variation back to an analytic curve��! .��/m ��k�m � S

whose closure passes throughx, and take the LMHS of that. The resulting. sH/x is independent of the choice of curve (up to the action of local mon-odromy mentioned earlier). In particular, lettingfNig denote the local mon-odromy logarithms, the weight filtrationM� on . sH/x is just the weight mon-odromy filtration attached to their sumN WD

P

aiNi (where thefaig are arbi-trary positive integers).

Now let r 2 N.

DEFINITION 26. A .higher/ normal functionoverS is a VMHS of the formV

in (the short-exact sequence)

0!H� V� ZS .0/! 0 (2-8)

whereH is a [pure] VHS of weight.�r/ and the [trivial, constant] variationZS .0/ has trivial monodromy. (The terminology “higher” only applies whenr > 1.) This is equivalent to a holomorphic, horizontal section of the generalizedJacobian bundle

J.H/ WDH

F0HCHZ

:

EXAMPLE 27. Given a smooth proper familyX�!S , with x0 2S . A higher al-

gebraic cycleZ2CH p.X ; r�1/prim WDkerfCH p.X ; r�1/!CH p.Xx0; r�1/

! Hgp;r�1.Xx0/g yields a section ofJ.R2p�r��C˝OS /DW J

p;r�1; this iswhat we shall mean by a(higher) normal function of geometric origin.7 (Thenotion ofmotivation overK likewise has an obvious extension from the classical1-parameter case in~ 1.)

7Note that Hgp;r �1.Xx0/Q WD H 2p�r C1.Xx0

;Q.p//\ F pH 2p�r C1.Xx0;C/ is actually zero

for r > 1, so that the “prim” comes for free for some multiple ofZ.


We now give the definition of admissibility for VMHS of the form in Def-inition 26 (but simplifying toD D fs1 � � � sk D 0g), starting with the localunipotent case. For this we need Deligne’s definition [De1] of the Ip;q.H /

of a MHSH , for which the reader may refer to Theorem 68 (in~ 4) below. Tosimplify notation, we shall abbreviateIp;q.H / to H .p;q/, so that, for instance,H.p;p/Q D Ip;p.H /\HQ, and drop the subscriptx for the LMHS notation.

DEFINITION 28. LetS D .��/k , V 2 NFr .S;H/Q (i.e., as in Definition 26,˝Q), andx D .0/.

(I) [unipotent case] Assume the monodromiesTi of H are unipotent, so that thelogarithmsNi and associated monodromy weight filtrationsM .i/

�are defined.

(Note that thefNig resp.fTig automatically commute, since any local systemmust be a representation of�1..�

�/k/, an abelian group.) We may “untwist”the local system Q via

QV WD exp

�

�1

2�p�1

X

i

log.si/Ni

�

V.Q/;

and setVe WD QV˝O�k for the Deligne extension. ThenV is ( NS-)admissible ifand only if

(a)H is polarizable,

(b) there exists a lift�Q 2 . QV/0 of 12Q.0/ such thatNi�Q 2M.i/�2. sH/Q (8i),

and

(c) there exists a lift�F .s/2� .S ;Ve/ of 12QS .0/ such that�F jS 2� .S;F0/.

(II) In general there exists a minimal finite cover� W .��/k ! .��/k (sendings ‘ s�) such that theT �i

i are unipotent.V is admissible if and only if��V

satisfies (a), (b), and (c).

The main result [K; SZ] is then thatV 2 NFr .S;H/adNS

has well-defined sV ,given as follows. On the underlying rational structure. QV/0 we put the weightfiltration Mi DMi sHCQ

˝

�Q

˛

for i � 0 andMi DMi sH for i < 0; whileon its complexification (Š .Ve/0) we put the Hodge filtrationF j DF j sHCC

C h�F .0/i for j � 0 and F j D F j sH for j > 0. (Here we are using theinclusion QH � QV, and the content of the statement is that this actually doesdefine a MHS.)

We can draw some further conclusions from (a)–(c) in case (I). With somework, it follows from (c) that

(c0) �F .0/ gives a lift of1 2Q.0/ satisfyingNi�F .0/ 2 . sH/.�1;�1/I

and one can also show thatNi�Q2M�2. sH/Q (8i). Furthermore, ifrD1 theneachNi�Q [resp.Ni�F .0/] belongs to the image underNi W sH! sH.�1/


of a rational [resp. type-.0; 0/] element. To see this, use the properties ofNi

to deduce that im.Ni/ � M.i/�r�1

; then for r D 1 we have, from (b) and (c),Ni�F .0/; Ni�Q 2M

.i/�2

.

(III) The definition of admissibility over an arbitrary smooth baseS togetherwith good compactificationNS is then local, i.e., reduces to the.��/k setting.Another piece of motivation for the definition of admissibility is this, for whichwe refer the reader to [BZ, Theorem 7.3]:

THEOREM 29. Any(higher) normal function of geometric origin is admissible.

2.12. Limits and singularities of ANFs. Now the idea of the “limit of anormal function” should be to interpret sV as an extension ofQ.0/ by sH.The obstruction to being able to do this is the singularity, as we now explain.All MHS in this section areQ-MHS.

According to [BFNP, Corollary 2.9], we have

NFr .S;H/adNS˝QŠ Ext1

VMHS.S/adNS

.Q.0/;H/;

as well as an equivalence of categories VMHS.S/adNS 'MHM.S/

psNS . We want to

push (in a sense canonically extend) our ANFV into NS and restrict the result tox.Of course, writing| WSŒ NS , |� is not right exact; so to preserve our extension,we take the derived functorR|� and land in the derived categoryDbMHM. NS/.Pulling back toDbMHM.fxg/ŠDbMHS by {�

x , we have defined an invariant.{�

xR|�/Hdg:

HomMHS

�

Q.0/;H 1K��

NFr .S;H/

singx

44

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

.{�xR|�/

Hdg// Ext1

DbMHS

�

Q.0/;K� WD {�xRj�H

�

OO

ker.singx/?

ffl

OO

limx// Ext1

MHS

�

Q.0/;H 0K��

OO

(2-9)

where the diagram makes a clear analogy to (2-1).For S D .��/k andHZ unipotent we have

K� '

˚

sH˚Ni�

M

i

sH.�1/�M

i<j

sH.�2/��

;

and the map

singx W NFr ..��/k ;H/ad�k ! .H 1

K�/.0;0/Q .Š coker.N /.�1/ for k D 1/


is induced byV‘ fNi�Qg � fNi�F .0/g. The limits, which are computed by

limx W ker.singx/! J�T

i ker.Ni/�

;

more directly generalize the 1-parameter picture. The target J.T

ker.Ni// isexactly what to put in over0 to get the multivariable pre-Neron-model.

We have introduced the general caser � 1 because of interesting applica-tions of higher normal functions to irrationality proofs, local mirror symmetry[DK]. In caser D 1 — we are dealing with classical normal functions — we canreplaceR|� in the above by perverse intermediate extension|!� (which by alemma in [BFNP] preserves the extension in this case: see Theorem 46 below).Correspondingly,K� is replaced by the local intersection cohomology complex

K�

red'˚

sH˚Ni�

M

i

Im.Ni/.�1/�M

i<j

Im.NiNj /.�2/! � � �

I

while the target for limx is unchanged, the one for singx is reduced to0 if kD 1

and to�

ker.N1/\ im.N2/

N2.kerN1/

�.�1;�1/

Q

(2-10)

if k D 2.

2.13. Applications of singularities. We hint at some good things to come:

(i) Replacing the singx-target (e.g., (2-10)) by actualimagesof ANFs, and usingtheir differences to glue pre-Neron components together yields a generalizedNeron model (over�r , or NS more generally) graphing ANFs. Again overx onegets an extension of a discrete (but not necessarily finite) singularity group bythe torusJ.\ ker.Ni//. A. Young [Yo] did this for abelian varieties, then [BPS]for general VHS. This will be described more precisely in~ 5.2.

(ii) (Griffiths and Green [GG]) The Hodge conjecture (HC) on a2p-dimensionalsmooth projective varietyX is equivalent to the following statement foreachprimitive Hodge.p;p/ class� and very ample line bundleL!X : there existsk� 0 such that the natural normal function8 �� overjLk jn OX (the complementof the dual variety in the linear system) has a nontorsion singularity at somepoint of OX . So, in asense, the analog of HC for.��/k is surjectivity of singxonto.H 1K�

red/.0;0/Q , and thisfails:

(iii) (M. Saito [S6], Pearlstein [Pe3]) LetH0=�� be a VHS of weight3 rank4

with nontrivial Yukawa coupling. Twisting it into weight�1, assume the LMHSis of typeII1: N 2 D 0, with GrM

�2 of rank 1. Take forH=.��/2 the pullbackof H0 by .s; t/ ‘ st . Then (2-10)¤ f0g D sing0fNF1..��/2;H/ad

�2g. The

8cf. ~ 3.2–3, especially (3-5).


obstruction to the existence of normal functions with nontrivial singularity isanalytic; and comes from a differential equation produced by the horizontalitycondition (see~ 5.4–5).

(iv) One can explain the meaning of the residue of the limitK1 class in Example25 above: writing|1 W .��/2Œ��, |2 W��Œ�2, factor.{�

xRj!�/Hdg

by .{�xR|2

� /Hdgı.{�

��|1!�/Hdg (where the{�Rj 2

� corresponds to the residue). Thatis, limit a normal function (or family of cycles) to a higher normal function (orfamily of higher Chow cycles) over a codimension-1 boundary component; thelatter can then have (unlike normal functions) a singularity in codimension1 —i.e., in codimension2 with respect to the original normal function.

This technique gives a quick proof of the existence of singularities for theCeresa cycle by limiting it to an Eisenstein symbol (see [Co]and the Introductionto [DK]). Additionally, one gets a geometric explanation ofwhy one does notexpect the singularities in (ii) to be supported in high-codimension substrata ofOX (supporting very degenerate hypersurfaces ofX ): along these substrata one

may reach (in the sense of (iv)) higher Chow cycles with rigidAJ invariants,hence no residues. For this reason codimension2 tends to be a better placeto look for singularities than in much higher codimension. These “shallow”substrata correspond to hypersurfaces with ordinary double points, and it wasthe original sense of [GG] that such points should trace out an algebraic cycle“dual” to the original Hodge class, giving aneffectiveproof of the HC.

3. Normal functions and the Hodge conjecture

In this section, we discuss the connection between normal functions and theHodge conjecture, picking up where~ 1 left off. We begin with a review ofsome properties of the Abel–Jacobi map. Unless otherwise noted, all varietiesare defined overC.

3.1. Zucker’s Theorem on Normal Functions.Let X be a smooth projectivevariety of dimensiondX . Recall thatJ p

h.X / is the intermediate Jacobian asso-

ciated to the maximal rationally defined Hodge substructureH of H 2p�1.X /

such thatHC �H p;p�1.X /˚H p�1;p.X /, and that (by a result of Lieberman[Li])

J p.X /algD im˚

AJX W CH p.X /alg! J p.X /

is a subabelian variety ofJ p.X /h:(3-1)

NOTATION 30. If f W X ! Y is a projective morphism thenf sm denotesthe restriction off to the largest Zariski open subset ofY over whichf issmooth. Also, unless otherwise noted, in this section, the underlying latticeHZ of every variation of Hodge structure is assumed to be torsion free, and


hence for a geometric familyf W X ! Y , we are really consideringHZ D

.Rkf sm�

Z/=ftorsiong.

As reviewed in~ 1, Lefschetz proved that every integral.1; 1/ class on a smoothprojective surface is algebraic by studying Poincare normal functions associ-ated to such cycles. We shall begin here by revisiting Griffiths’ program (alsorecalled in~ 1) to prove the Hodge conjecture for higher codimension classes byextending Lefschetz’s methods: By induction on dimension,the Hodge conjec-ture can be reduced to the case of middle-dimensional Hodge classes on even-dimensional varieties [Le1, Lecture 14]. Suppose therefore thatX � Pk is asmooth projective variety of dimension2m. Following [Zu2, ~ 4], let us picka Lefschetz pencil of hyperplane sections ofX , i.e., a family of hyperplanesHt � Pk of the formt0w0C t1w1 D 0 parametrized byt D Œt0; t1� 2 P1 relativeto a suitable choice of homogeneous coordinatesw D Œw0; : : : ; wk � on Pk suchthat:

� for all but finitely many pointst 2 P1, the corresponding hyperplane sectionof Xt DX \Ht is smooth;

� the base locusB DX \fw 2 Pk j w0 D w1 D 0g is smooth; and� each singular hyperplane section ofX has exactly one singular point, which

is an ordinary double point.

Given such a Lefschetz pencil, let

Y D f .x; t/ 2X �P1 j x 2Ht g

and let� W Y ! P1 denote projection onto the second factor. LetU denote theset of pointst 2 P1 such thatXt is smooth andH be the variation of Hodgestructure overU with integral structureHZDR2m�1�sm

�Z.m/. Furthermore, by

Schmid’s nilpotent orbit theorem [Sc], the Hodge bundlesF� have a canonicalextension to a system of holomorphic bundlesF�

e over P1. Accordingly, wehave a short exact sequence of sheaves

0! j�HZ!He=Fme ! J

me ! 0; (3-2)

wherej W U ! P1 is the inclusion map. As before, let us call an element� 2H 0.P1;J m

e / a Poincare normal function. Then, we have the following tworesults [Zu2, Thms. 4.57, 4.17], the second of which is knownas the Theoremon Normal Functions:

THEOREM 31. Every Poincare normal function satisfies Griffiths horizontality.

THEOREM 32. Every primitive integral Hodge class onX is the cohomologyclass of a Poincare normal function.


The next step in the proof of the Hodge conjecture via this approach is to showthat for t 2 U , the Abel–Jacobi map

AJ W CH m.Xt /hom! J m.Xt /

is surjective. However, form > 1 this is rarely true (even granting the con-jectural equality ofJ m.X /alg and J m

h.X /) sinceJ m.Xt / ¤ J m

h.Xt / unless

H 2m�1.Xt ;C/DH m;m�1.Xt /˚H m�1;m.Xt /. In plenty of cases of interestJ m

h.X / is in fact trivial; Theorem 33 and Example 35 below give two different

instances of this.

THEOREM 33 [Le1, Example 14.18].If X � Pk is a smooth projective varietyof dimension2m such thatH 2m�1.X / D 0 and fXtg is a Lefschetz pencil ofhyperplane sections ofX such thatFmC1H 2m�1.Xt / ¤ 0 for every smoothhyperplane section, then for generict 2 U , J m

h.Xt /D 0.

THEOREM 34. If Jp

h.X /D 0, then the image ofCH m.W /hom in J p.X / under

the Abel–Jacobi map is countable.

SKETCH OF PROOF. As a consequence of (3-1), ifJp

h.X /D 0 the Abel–Jacobi

map vanishes onCH p.X /alg. Therefore, the cardinality of the image of theAbel–Jacobi map onCH p.X /hom is bounded by the cardinality of the GriffithsgroupCH p.X /hom=CH p.X /alg, which is known to be countable. ˜

EXAMPLE 35. Specific hypersurfaces withJ p

h.X / D 0 were constructed by

Shioda [Sh]: LetZnm denote the hypersurface inPnC1 defined by the equation

nC1X

iD0

xixm�1iC1 D 0 .xnC2 D x0/:

Suppose thatnD 2p� 1 > 1, m� 2C 3=.p� 1/ and

d0 D f.m� 1/nC1C .�1/nC1g=m

is prime. ThenJ p

h.Zn

m/D 0.

3.2. Singularities of admissible normal functions. In [GG], Griffiths andGreen proposed an alternative program for proving the Hodgeconjecture bystudying the singularities of normal functions over higher-dimensional parame-ter spaces. Following [BFNP], letS a complex manifold andHD .HZ;F

�HO/

be a variation of polarizable Hodge structure of weight�1 over S . Then, wehave the short exact sequence

0!HZ!H=F0! J.H/! 0


of sheaves and hence an associated long exact sequence in cohomology. Inparticular, the cohomology class cl.�/ of a normal function� 2 H 0.S;J.H//

is just the image of� under the connecting homomorphism

@ WH 0.S;J.H//!H 1.S;HZ/:

Suppose now thatS is a Zariski open subset of a smooth projective varietyNS . Then the singularity of� at p 2 NS is the quantity

�Z;p.�/D lim��!p2U

cl.�jU \S / 2 lim��!p2U

H 1.U \S;HZ/D .R1j�HZ/p

where the limit is taken over all analytic open neighborhoods U of p, andj W

S! NS is the inclusion map. The image of�Z;p.�/ in cohomology with rationalcoefficients will be denoted by singp.��/.

REMARK 36. If p 2 S then�Z;p.�/D 0.

THEOREM 37 [S1]. Let � be an admissible normal function on a Zariski opensubset of a curveNS . Then, �Z;p.�/ is of finite order for each pointp 2 NS .

PROOF. By [S1], an admissible normal function� W S ! J.H/ is equivalent toan extension

0!H! V! Z.0/! 0 (3-3)

in the category of admissible variations of mixed Hodge structure. By the mon-odromy theorem for variations of pure Hodge structure, the local monodromy ofV about any pointp 2 NS �S is always quasi-unipotent. Without loss of gener-ality, let us assume that it is unipotent and thatT D eN is the local monodromyof V at p acting on some fixed reference fiber with integral structureVZ. Then,due to the length of the weight filtrationW , the existence of the relative weightfiltration of W andN is equivalent to the existence of anN -invariant splitting ofW [SZ, Proposition 2.16]. In particular, leteZ 2 VZ project to12GrW0 ŠZ.0/.Then, by admissibility, there exists an elementhQ 2HQ DW�1\VQ such that

N.eZC hQ/D 0

and hence.T � I/.eZ C hQ/ D 0.9 Any two such choices ofeZ differ by anelementhZ 2W�1\VZ. Therefore, an admissible normal function� determinesa class

Œ��D Œ.T � I/eZ� 2.T � I/.HQ/

.T � I/.HZ/

Tracing through the definitions, one finds that the left-handside of this equationcan be identified with�Z;p.�/, whereas the right-hand side is exactly the torsionsubgroup of.R1j�HZ/p. ˜

9Alternatively, one can just derive this from Definition 28(I).


DEFINITION 38 [BFNP]. An admissible normal function� defined on a Zariskiopen subset ofNS is singular on NS if there exists a pointp 2 NS such thatsingp.�/¤ 0.

Let S be a complex manifold andf WX ! S be a family of smooth projectivevarieties overS . Let H be the variation of pure Hodge structure of weight�1

over S with integral structureHZ D R2p�1f�Z.p/. Then, an elementw 2J p.X / .D J 0.H 2p�1.X;Z.p//// defines a normal function�w W S ! J.H/

by the rule�w.s/D i�

s .w/; (3-4)

whereis denotes inclusion of the fiberXsD f�1.s/ into X . More generally, let

H2pD.X;Z.p// denote the Deligne cohomology ofX , and recall that we have a

short exact sequence

0! J p.X /!H2pD.X;Z.p//!H p;p.X;Z.p//! 0:

Call a Hodge class

� 2H p;p.X;Z.p// WDH p;p.X;C/\H 2p.X;Z.p//

primitive with respect tof if i�s .�/ D 0 for all s 2 S , and letH p;p

prim.X;Z.p//

denote the group of all such primitive Hodge classes. Then, by the functorialityof Deligne cohomology, a choice of liftingQ� 2 H

2pD.X;Z.p// of a primitive

Hodge class� determines a map�Q�WS! J.H/. A short calculation (cf. [CMP,

Ch. 10]) shows that�Q�is a (horizontal) normal function overS . Furthermore,

in the algebraic setting (meaning thatX;S; f are algebraic),�Q�is an admissible

normal function [S1]. Let ANF.S;H/ denote the group of admissible normalfunctions with underlying variation of Hodge structureH. By abuse of notation,let J p.X /�ANF.S;H/ denote the image of the intermediate JacobianJ p.X /

in ANF.S;H/ under the mapw ‘ �w. Then, since any two liftsQ� of � toDeligne cohomology differ by an element of the intermediateJacobianJ p.X /,it follows that we have a well-defined map

AJ WHp;pprim.X;Z.p//! ANF.S;H/=J p.X /: (3-5)

REMARK 39. We are able to drop the notation NF.S;H/adNS

used in~ 2, becausein the global algebraic case it can be shown that admissibility is independent ofthe choice of compactificationNS .

3.3. The Main Theorem. Returning to the program of Griffiths and Green, letX be a smooth projective variety of dimension2m andL!X be a very ampleline bundle. Let NP D jLj and

X D˚

.x; s/ 2X � NP j s.x/D 0

(3-6)


be the incidence variety associated to the pair.X;L/. Let � W X ! NP denoteprojection on the second factor, and letOX � NP denote the dual variety ofX(the pointss 2 NP such thatXs D �

�1.s/ is singular). LetH be the variationof Hodge structure of weight�1 overP D NP � OX attached to the local systemR2m�1�sm

�Z.m/.

For a pair.X;L/ as above, an integral Hodge class� of type .m;m/ on X

is primitive with respect to�sm if and only if it is primitive in the usual senseof being annihilated by cup product withc1.L/. Let H m;m

prim .X;Z.m// denotethe group of all such primitive Hodge classes, and note thatH m;m

prim .X;Z.m// isunchanged upon replacingL by L˝d for d > 0. Given � 2 H m;m

prim .X;Z.m//,let

�� DAJ.�/ 2 ANF.P;H/=J m.X /

be the associated normal function (3-5).

LEMMA 40. If �w W P ! J.H/ is the normal function(3-4) associated to anelementw 2 J m.X / thensingp.�w/D 0 at every pointp 2 OX .

Accordingly, for any pointp 2 OX we have a well defined map

singp W ANF.P;H/=J m.X /! .R1j�HQ/p

which sends the elementŒ��2ANF.P;H/=J m.X / to singp.�/. In keeping with

our prior definition, we say that�� is singular on NP if there exists a pointp 2 OXsuch that singp.�/¤ 0.

CONJECTURE41 [GG; BFNP].Let L be a very ample line bundle on a smoothprojective varietyX of dimension2m. Then, for every nontorsion class� inH

m;mprim .X;Z.m// there exists an integerd > 0 such thatAJ.�/ is singular onNP D jL˝d j.

THEOREM 42 [GG; BFNP; dCM].Conjecture41holds( for every even-dimen-sional smooth projective variety) if and only if the Hodge conjecture is true.

To outline the proof of Theorem 42, observe that for any pointp 2 OX , we havethe diagram

Hm;mprim .X;Z.m//

AJ//

p

››

ANF.P;H/=J m.X /

singp››

H 2m.Xp;Q.m//p

??// .R1j�HQ/p

(3-7)

where p WHm;mprim .X;Z.m//!H 2m.Xp;Q.m// is the restriction map.


Suppose that there exists a map

p WH2m.Xp;Q.m//! .R1j�HQ/p; (3-8)

which makes the diagram (3-7) commute, and that after replacing L by L˝d forsomed > 0 the restriction of p to the image of p is injective. Then, existenceof a pointp 2 OX such that singp.��/¤ 0 implies that the Hodge class� restrictsnontrivially toXp. Now recall that by Poincare duality and the Hodge–Riemannbilinear relations, the Hodge conjecture for a smooth projective varietyY isequivalent to the statement that for every rational.q; q/ class onY there existsan algebraic cycleW of dimension2q on Y such that [ ŒW �¤ 0.

Let f W QXp ! Xp be a resolution of singularities ofXp and g D i ı f ,wherei W Xp ! X is the inclusion map. By a weight argumentg�.�/ ¤ 0,and so there exists a class� 2 Hgm�1. QXp/ with � [ � ¤ 0. Embedding QXp insome projective space, and inducing onevendimension, we can assume thatthe Hodge conjecture holds for a general hyperplane sectionI W YŒ QXp. Thisyields an algebraic cycleW on Y with ŒW � D I�.�/. Varying Y in a pencil,and using weak Lefschetz,W traces out10 a cycleW D

P

j aj Wj on QXp withŒW �D �, so thatg�.�/[ ŒW �¤ 0; in particular,� [g�ŒWj �¤ 0 for somej .

Conversely, by the work of Thomas [Th], if the Hodge conjecture is true thenthe Hodge class� must restrict nontrivially to some singular hyperplane sectionof X (again for someL˝d for d sufficiently large). Now one uses the injectivityof p on im. p/ to conclude that�� has a singularity.

EXAMPLE 43. Let X � P3 be a smooth projective surface. For every� 2H

1;1prim.X;Z.1//, there is a reducible hypersurface sectionXp � X and compo-

nent curveW of Xp such that deg.�jW / ¤ 0. (Note that deg.�jXp/ is neces-

sarily 0.) As the reader should check, this follows easily from Lefschetz (1,1).Moreover (writingd for the degree ofXp), p is a point in a codimension� 2

substratumS 0 of OX �PH 0.O.d// (since fibers over codimension-one substrataare irreducible), and singq.��/¤ 0 8q 2 S 0.

REMARK 44. There is a central geometric issue lurking in Conjecture41:

If the HC holds, andL D OX .1/ (for some projective embedding ofX ), isthere some minimumd0 — uniform in some sense — for whichd � d0 impliesthat�� is singular?

In [GG] it is established that, at best, such ad0 could only be uniform inmoduli of the pair.X; �/. (For example, in the case dim.X / D 2, d0 is of theform C �j� ��j, for C a constant. Since the self-intersection numbers of integral

10More precisely, one uses here a spread or Hilbert scheme argument. See for example the beginning ofChapter 14 of [Le1].


classes becoming Hodge in various Noether–Lefschetz loci increase withoutbound, there is certainly not anyd0 uniform in moduli ofX .) Whether thereis some such “lower bound” of this form remains an open question in higherdimension.

3.4. Normal functions and intersection cohomology.The construction of themap p depends on the decomposition theorem of Beilinson, Bernstein, andDeligne [BBD] and Morihiko Saito’s theory of mixed Hodge modules [S4]. Asfirst step in this direction, recall [CKS2] that ifH is a variation of pure Hodgestructure of weightk defined on the complementSD NS�D of a normal crossingdivisor on a smooth projective varietyNS then

H `.2/.S;HR/Š IH`. NS ;HR/;

where the left-hand side isL2-cohomology and the right-hand side is inter-section cohomology. Furthermore, via this isomorphismIH`. NS ;HC/ inherits acanonical Hodge structure of weightkC `.

REMARK 45. If Y is a complex algebraic variety, MHM.Y / is the category ofmixed Hodge modules onY . The category MHM.Y / comes equipped with afunctor

rat WMHM.Y /! Perv.Y /

to the category of perverse sheaves onY . If Y is smooth andV is a variation ofmixed Hodge structure onY thenV ŒdY � is a mixed Hodge module onY , andrat.V ŒdY �/ Š VŒdY � is just the underlying local systemV shifted into degree�dY .

If Y ı is a Zariski open subset ofY andP is a perverse sheaf onY ı then

IH`.Y;P/DH`�dY .Y; j!�P ŒdY �/

wherej!� is the middle extension functor [BBD] associated to the inclusion mapj W Y ı! Y . Likewise, for any pointy 2 Y , the local intersection cohomologyof P at y is defined to be

IH`.Y;P/y DHk�dY .fyg; i�j!�P ŒdY �/

wherei W fyg! Y is the inclusion map. IfP underlies a mixed Hodge module,the theory of MHM puts natural MHS on these groups, which in particular ishow the pure HS onIH`. NS ;HC/ comes about.

THEOREM46 [BFNP, Theorem 2.11].Let NS be a smooth projective variety andH be a variation of pure Hodge structure of weight�1 on a Zariski open subsetS � NS . Then, the group homomorphism

cl W ANF.S;H/!H 1.S;HQ/


factors through IH1. NS ;HQ/.

SKETCH OF PROOF. Let � 2 ANF.S;H/ be represented by an extension

0!H! V! Z.0/! 0

in the category of admissible variations of mixed Hodge structure onS . Letj W S ! NS be the inclusion map. Then, becauseV has only two nontrivialweight graded quotients which are adjacent, it follows by [BFNP, Lemma 2.18]that

0! j!�HŒdS �! j!�V ŒdS �!Q.0/ŒdS �! 0

is exact in MHM. NS/. ˜

REMARK 47. In this particular context,j!�V ŒdS � can be described as the uniqueprolongation ofV ŒdS � to NS with no nontrivial sub or quotient object supportedon the essential image of the functori WMHM.Z/!MHM. NS/whereZD NS�S

andi WZ! NS is the inclusion map.

In the local case of an admissible normal function on a product of puncturedpolydisks.��/r with unipotent monodromy, the fact that sing0.�/ (where0 isthe origin of�r � .��/r ) factors through the local intersection cohomologygroups can be seen as follows: Such a normal function� gives a short exactsequence of local systems

0!HQ! VQ!Q.0/! 0

over .��/r . Fix a reference fiberVQ of VQ and letNj 2 Hom.VQ;VQ/ denotethe monodromy logarithm ofVQ about thej -th punctured disk. Then [CKS2],we get a complex of finite-dimensional vector spaces

Bp.VQ/DM

i1<i2<��<ip

Ni1Ni2� � �Nip .VQ/

with differentiald , which acts on the summands ofBp.VQ/ by the rule

Ni1� � � ONi`

� � �NipC1.VQ/

.�1/`�1Ni`� Ni1

� � �Ni`� � �NipC1

.VQ/

(and taking the sum over all insertions). LetB�.HQ/ andB�.Q.0// denote theanalogous complexes attached to the local systemsHQ andQ.0/. By [GGM],the cohomology of the complexB�.HQ/ computes the local intersection coho-mology of HQ. In particular, since the complexesB�.Q.0// andB�.HQ/ sitinside the standard Koszul complexes which compute the ordinary cohomologyof Q.0/ andHQ, in order show that sing0 factors throughIH1.HQ/ it is sufficient


to show that@cl.�/2H 1..��/r ;HQ/ is representable by an element ofB1.HQ/.Indeed, letv be an element ofVQ which maps to1 2Q.0/. Then,

@ cl.�/D @1D Œ.N1.v/; : : : ;Nr .v//�

By admissibility and the short length of the weight filtration, for eachj thereexists an elementhj 2 HQ such thatNj .hj / D Nj .v/, which is exactly thecondition that

.N1.v/; : : : ;Nr .v// 2B1.VQ/:

THEOREM48 [BFNP, Theorem 2.11].Under the hypothesis of Theorem46, forany pointp 2 NS the group homomorphismsingp W ANF.S;H/! .R1j�HQ/pfactors through the local intersection cohomology group IH1.HQ/p.

To continue, we need to pass from Deligne cohomology to absolute Hodge coho-mology. Recall that MHM.Spec.C// is the category MHS of graded-polarizableQ mixed Hodge structures. LetQ.p/ denote the Tate object of type.�p;�p/ inMHS andQY .p/Da�

YQ.p/whereaY WY !Spec.C/ is the structure morphism.

Let QY DQY .0/.

DEFINITION 49. LetM be an object of MHM.Y /. Then,

H nAH.Y;M /D HomDbMHM .QY ;M Œn�/

is the absolute Hodge cohomology ofM .

The functor ratWMHM.Y /! Perv.Y / induces a “cycle class map”

rat WH nAH.Y;M /!H

n.Y; rat.M //

from the absolute Hodge cohomology ofM to the hypercohomology of rat.M /.In the case whereY is smooth and projective,H 2p

AH.Y;QY .p// is the Deligne

cohomology groupH 2pD.Y;Q.p// and rat is the cycle class map on Deligne

cohomology.

DEFINITION 50. Let NS be a smooth projective variety andV be an admissiblevariation of mixed Hodge structure on a Zariski open subsetS of NS . Then,

IHnAH.NS ;V/D HomDbMHM. NS/.Q NS ŒdS � n�; j!�V ŒdS �/;

IHnAH.NS ;V/s D HomDbMHS.QŒdS � n�; i�j!�V ŒdS �/;

wherej W S ! NS andi W fsg ! NS are inclusion maps.

The following lemma links absolute Hodge cohomology and admissible normalfunctions:

LEMMA 51. [BFNP, Proposition 3.3]LetH be a variation of pure Hodge struc-ture of weight�1 defined on a Zariski open subsetS of a smooth projectivevariety NS . Then, IH1

AH. NS ;H/Š ANF.S;H/˝Q.


3.5. Completion of the diagram (3-7). Let f W X ! Y be a projective mor-phism between smooth algebraic varieties. Then, by the workof Morihiko Saito[S4], there is a direct sum decomposition

f�QX ŒdX �DM

i

H i .f�QX ŒdX �/ Œ�i � (3-9)

in MHM.Y /. Furthermore, each summandH i.f�QX ŒdX �/ is pure of weightdX C i and admits a decomposition according to codimension of support:

H i .f�QX ŒdX �/ Œ�i �DL

j Eij Œ�i �I (3-10)

i.e., Eij Œ�i � is a sum of Hodge modules supported on codimensionj subva-rieties ofY . Accordingly, we have a system of projection operators (insertingarbitrary twists)

L

˘ij WHnAH.X;Q.`/ŒdX �/

Š

!L

ij H n�iAH

.Y;Eij .`//;

L

˘ij WHnAH.Xp;Q.`/ŒdX �/

Š

!L

ij H n�iAH

.Y; ��Eij .`//;

L

˘ij WHn.X; rat.Q.`/ŒdX �//

Š

!L

ij Hn�i.Y; rat.Eij .`///;

L

˘ij WHn.Xp; rat.Q.`/ŒdX �//

Š

!L

ij Hn�i.Y; ��rat.Eij .`///;

wherep 2 Y and� W fpg ! Y is the inclusion map.

LEMMA 52 [BFNP, Equation 4.12].Let Hq D Rqf sm� QX and recall that we

have a decomposition

H2k�1 DH

2k�1van ˚H

2k�1fix

whereH2k�1fix is constant andH2k�1

van has no global sections. For any pointp 2 Y , we have a commutative diagram

H 2kAH.X;Q.k//

i�

››

˘// ANF.Y sm;H2k�1

van .k//

i�

››

H 2k.Xp;Q.k// ˘// IH1.H2k�1.k//p

(3-11)

whereY sm is the largest Zariski open set over whichf is smooth and isinduced by r0 for r D 2k � 1� dX C dY .

We now return to the setting of Conjecture 41:X is a smooth projective varietyof dimension2m, L is a very ample line bundle onX andX is the associatedincidence variety (3-6), with projections� WX ! NP and prWX !X . Then, wehave the following “Perverse weak Lefschetz theorem”:


THEOREM 53 [BFNP, Theorem 5.1].LetX be the incidence variety associatedto the pair.X;L/ and��QX D

L

ij Eij in accord with(3-9)and(3-10).Then:

(i) Eij D 0 unlessi � j D 0.(ii) Ei0 DH i.X;QX Œ2m� 1�/˝Q NP Œd NP �. for i < 0.

Note that by hard Lefschetz,Eij ŠE�i;j .�i/ [S4].To continue, recall that given a Lefschetz pencil� � NP of hyperplane sec-

tions of X , we have an associated system of vanishing cyclesfıpgp2�\ OX �

H 2m�1.Xt ;Q/ on the cohomology of the smooth hyperplane sectionsXt ofX with respect to�. As one would expect, the vanishing cycles of� arenonvanishingif for some (hence all)p 2 �\ OX , ıp ¤ 0 (in H 2m�1.Xt ;Q/).Furthermore, this property depends only onL and not the particular choice ofLefschetz pencil�. This property can always be arranged by replacingL byL˝d for somed > 0.

THEOREM 54. If all vanishing cycles are nonvanishing thenE01 D 0. Other-wise, E01 is supported on a dense open subset ofOX .

Using the Theorems 53 and 54, we now prove that the diagram

H 2mD.X;Z.m//prim

AJ//

pr�

››

ANF.P;H/=J m.X /

˝Q

››

H 2mAH.X ;Q.m//

˘// ANF.P;Hvan/˝Q

(3-12)

commutes, whereH 2mD.X;Z.m//prim is the subgroup ofH 2m

D.X;Z.m// whose

elements project to primitive Hodge classes inH 2m.X;Z.m//, and˘ is in-duced by 00 together with projection ontoHvan. Indeed, by the decompositiontheorem,

H 2mAH.X ;Q.m//DH

1�d NP

AH.X ;Q.m/Œ2mC d NP � 1�/

DL

H1�d NP

AH. NP ;Eij .m/Œ�i �/:

Let Q� 2H 2mD.X;Z.m// be a primitive Deligne class and!D

L

ij !ij denote

the component of! D pr�. Q�/ with respect toEij .m/Œ�i � in accord with theprevious equation. Then, in order to prove the commutativity of (3-12) it issufficient to show that.!/q D .!00/q for all q 2 P . By Theorem 53, we knowthat!ij D0 unlessij D0. Furthermore, by [BFNP, Lemma 5.5],.!0j /qD0 forj > 1. Likewise, by Theorem 54,.!01/q D 0 for q 2 P sinceE01 is supportedon OX .

Thus, in order to prove the commutativity of (3-12), it is sufficient to show that.!i0/q D 0 for i > 0. However, as a consequence of Theorem 53(ii),Ei0.m/D


KŒd NP �, whereK is a constant variation of Hodge structure onNP ; and hence

H 1�d NP .X ;Ei0.m/Œ�i �/D Ext1�d NP

DbMHM. NP/.Q NP ;KŒd NP � i �/

D Ext1�i

DbMHM. NP/.Q NP ;K/:

Therefore,.!i0/q D 0 for i > 1 while .!10/q corresponds to an element ofHom.Q.0/;Kq/ whereK is the constant variation of Hodge structure withfiber H 2m.Xq;Q.m// over q 2 P . It therefore follows from the fact thatQ� isprimitive that .!10/q D 0. Splicing diagram (3-12) together with (3-11) (andreplacingf WX ! Y by � W X ! NP , etc.) now gives the diagram (3-7).

REMARK 55. The effect of passing fromH toHvan in the constructions above isto annihilateJ m.X /�H 2m

D.X;Z.m//prim. Therefore, in (3-12) we can replace

H 2mD.X;Z.m//prim by H

m;mprim .X;Z.m//.

Finally, if all the vanishing cycles are nonvanishing,E01 D 0. Using this fact,we then get the injectivity ofp on the image of p.

Returning to the beginning of this section, we now see that although extend-ing normal functions along Lefschetz pencils is insufficient to prove the Hodgeconjecture for higher codimension cycles, the Hodge conjecture is equivalent toa statement about the behavior of normal functions on the complement of thedual variety ofX insidejLj for L� 0. We remark that an interpretation of theGHC along similar lines has been done recently by the authorsin [KP].

4. Zeroes of normal functions

4.1. Algebraicity of the zero locus. Some of the deepest evidence to date insupport of the Hodge conjecture is the following result of Cattani, Deligne andKaplan on the algebraicity of the Hodge locus:

THEOREM 56 [CDK]. LetH be a variation of pure Hodge structure of weight0

over a smooth complex algebraic varietyS . Let˛sobe an integral Hodge class

of type.0; 0/ on the fiber ofH at so. Let U be a simply connected open subsetof S containingso and˛ be the section ofHZ overU defined by parallel trans-lation of ˛so

. Let T be the locus of points inU such that .s/ is of type.0; 0/on the fiber ofH overs. Then, the analytic germ ofT at p is the restriction ofa complex algebraic subvariety ofS .

More precisely, as explained in the introduction of [CDK], in the case whereH arises from the cohomology of a family of smooth projective varietiesf WX ! S , the algebraicity of the germ ofT follows from the Hodge conjecture.A natural analog of this result for normal functions is this:


THEOREM 57. Let S be a smooth complex algebraic variety, and � W S !

J.H/ be an admissible normal function, whereH is a variation of pure Hodgestructure of weight�1. Then, the zero locus

Z.�/D f s 2 S j �.s/D 0 g

is a complex algebraic subvariety ofS .

This theorem was still a conjecture when the present articlewas submitted, andhas just been proved by the second author in work with P. Brosnan [BP3]. It is ofparticular relevance to the Hodge conjecture, due to the following relationshipbetween the algebraicity ofZ.�/ and the existence of singularities of normalfunctions. Say dim.X / D 2m, and let.X;L; �/ be a triple consisting of asmooth complex projective varietyX , a very ample line bundleL on X anda primitive integral Hodge class� of type .m;m/. Let �� (assumed nonzero)

be the associated normal function on the complement of the dual variety OXconstructed in~ 3, andZ be its zero locus. Then, assuming thatZ is algebraicand positive-dimensional, the second author conjectured that� should have sin-gularities along the intersection of the closure ofZ with OX .

THEOREM 58 [Sl1]. Let .X;L; �/ be a triple as above, and assume thatL issufficiently ample that, given any pointp 2 OX , the restriction of p to the imageof p in diagram(3-7) is injective. Suppose thatZ contains an algebraic curve.Then, �� has a nontorsion singularity at some point of the intersection of the

closure of this curve withOX .

SKETCH OF PROOF. LetC be the normalization of the closure of the curve inZ.Let X ! NP be the universal family of hyperplane sections ofX over NP D jLjandW be the pullback ofX to C . Let � WW ! C be the projection map, andU the set of pointsc 2 C such that��1.c/ is smooth andWU D �

�1.U /. Viathe Leray spectral sequence for�, it follows that restriction of� to WU is zerobecauseU �Z and� is primitive. On the other hand, sinceW “ X is finite, �must restrict (pull back) nontrivially toW , and hence� must restrict nontriviallyto the fiber��1.c/ for some pointc 2 C in the complement ofU . ˜

Unfortunately, crude estimates for the expected dimensionof the zero locusZarising in this context appear to be negative. For instance,take X to be anabelian surface in the following:

THEOREM 59. LetX be a surface andLDOX .D/ be an ample line bundle onX . Then, for n sufficiently large, the expected dimension of the zero locus of thenormal function�� attached to the triple.X;L˝n; �/ as above is

h2;0�h1;0 � n.D:K/� 1;

whereK is the canonical divisor ofX .


SKETCH OF PROOF. Since Griffiths’ horizontality is trivial in this setting,com-puting the expected dimension boils down to computing the dimension ofjLjand genus of a smooth hyperplane section ofX with respect toL. ˜

REMARK 60. In Theorem 59, we construct�� from a choice of lift to Delignecohomology (or an algebraic cycle) to get an element of ANF.P;H/. But this isdisingenuous, since we are starting with a Hodge class. It ismore consistent towork with �� 2ANF.P;H/=J 1.X / as in equation (3-5), and then the dimensionestimate improves by dim.J 1.X // D h1;0 to h2;0 � n.D:K/ � 1. Notice thatthis salvages at least the abelian surface case (though it isstill a crude estimate).For surfaces of general type, one is still in trouble withoutmore information,like the constantC in Remark 44.

We will not attempt to describe the proof of Theorem 57 in general, but we willexplain the following special case:

THEOREM61 [BP2].LetS be a smooth complex algebraic variety which admitsa projective completionNS such thatD D NS �S is a smooth divisor. LetH be avariation of pure Hodge structure of weight�1 on S and� W S ! J.H/ be anadmissible normal function. Then, the zero locusZ of � is an complex algebraicsubvariety ofS .

REMARK 62. This result was obtained contemporaneously by MorihikoSaitoin [S5].

In analogy with the proof of Theorem 56 on the algebraicity ofthe Hodge lo-cus, which depends heavily on the several variable SL2-orbit theorem for nilpo-tent orbits of pure Hodge structure [CKS1], the proof of Theorem 57 dependsupon the corresponding result for nilpotent orbits of mixedHodge structure. Forsimplicity of exposition, we will now review the1-variable SL2-orbit theoremin the pure case (which is due to Schmid [Sc]) and a version of the SL2-orbittheorem in the mixed case [Pe2] sufficient to prove Theorem 61. For the proof ofTheorem 57, we need the full strength of the several variableSL2-orbit theoremof Kato, Nakayama and Usui [KNU1].

4.2. The classical nilpotent and SL2-orbit theorems. To outline the proof ofTheorem 61, we now recall the theory of degenerations of Hodge structure: LetH be a variation of pure Hodge structure of weightk over a simply connectedcomplex manifoldS . Then, via parallel translation back to a fixed referencefiber H DHso

we obtain a period map

' W S !D; (4-1)

whereD is Griffiths’ classifying space of pure Hodge structures onH with fixedHodge numbersfhp;k�pgwhich are polarized by the bilinear formQ of H . The


setD is a complex manifold upon which the Lie group

GR D AutR.Q/

acts transitively by biholomorphisms, and henceD Š GR=GFo

R , whereGFo

R isthe isotropy group ofFo 2D. The compact dual ofD is the complex manifold

LD ŠGC=GFo

C

whereFo is any point inD. (In general,F DF � denotes a Hodge filtration.) IfS is not simply connected, then the period map (4-1) is replaced by

' W S ! � nD (4-2)

where� is the monodromy group ofH! S acting on the reference fiberH .For variations of Hodge structure of geometric origin,S will typically be a

Zariski open subset of a smooth projective varietyNS . By Hironaka’s resolutionof singularities theorem, we can assumeDD NS �S to be a divisor with normalcrossings. The period map (4-2) will then have singularities at the points ofDabout whichH has nontrivial local monodromy. A precise local description ofthe singularities of the period map of a variation of Hodge structure was obtainedby Schmid [Sc]: Let' W .��/r ! � nD be the period map of variation of purepolarized Hodge structure over the product of punctured disks. First, one knowsthat ' is locally liftable with quasi-unipotent monodromy. Afterpassage to afinite cover, we therefore obtain a commutative diagram

U rF

//

››

D

››

.��/r'

// � nD

(4-3)

whereU r is the r -fold product of upper half-planes andU r ! .��/r is thecovering map

sj D e2� izj ; j D 1; : : : ; r

with respect to the standard Euclidean coordinates.z1; : : : ; zr / onU r �Cr and.s1; : : : ; sr / on .��/r � Cr .

Let Tj D eNj denote the monodromy ofH aboutsj D 0. Then,

.z1; : : : ; zr /D e�P

j zj Nj :F.z1; : : : ; zr /

is a holomorphic map fromU r into LD which is invariant under the transforma-tion zj ‘ zj C 1 for eachj , and hence drops to a map.��/r ! LD which wecontinue to denote by .


DEFINITION 63. LetD be a classifying space of pure Hodge structure withassociated Lie groupGR. LetgR be the Lie algebra ofGR. Then, a holomorphic,horizontal map� W Cr ! LD is a nilpotent orbit if

(a) there exists > 0 such that�.z1; : : : ; zr / 2D if Im .zj / > ˛ 8j ; and(b) there exist commuting nilpotent endomorphismsN1; : : : ;Nr 2 gR and a

point F 2 LD such that�.z1; : : : ; zr /D eP

j zj Nj :F .

THEOREM 64 (NILPOTENT ORBIT THEOREM [Sc]). Let' W .��/r ! � nD bethe period map of a variation of pure Hodge structure of weight k with unipotentmonodromy. Let dD be aGR-invariant distance onD. Then:

(a)F1 D lims!0 .s/ exists, i.e., .s/ extends to a map�r ! LD;(b) �.z1; : : : ; zr /D e

P

j zj Nj :F1 is a nilpotent orbit; and(c) there exist constantsC , ˛ andˇ1; : : : ; ˇr such that ifIm.zj / > ˛ 8j then

�.z1; : : : ; zr / 2D and

dD.�.z1; : : : ; zr /;F.z1; : : : ; zr // < CX

j

Im.zj / j e�2� Im.zj /:

REMARK 65. Another way of stating part (a) of this theorem is that theHodgebundlesFp of HO extend to a system of holomorphic subbundles of the canon-ical extension ofHO. Indeed, recall from~ 2.7 that one way of constructing amodel of the canonical extension in the unipotent monodromycase is to take aflat, multivalued framef�1; : : : ; �mg of HZ and twist it to form a single valuedholomorphic framef Q�1; : : : ; Q�mg over .��/r where Q�j D e�

12�i

P

j log.sj /Nj �j ,and then declaring this twisted frame to define the canonicalextension.

Let N be a nilpotent endomorphism of a finite-dimensional vector space over afield k. Then,N can be put into Jordan canonical form, and hence (by consider-ing a Jordan block) it follows that there is a unique, increasing filtration W.N /

of V , such that, for each indexj ,

(a) N.W.N /j /�W.N /j�2 and

(b) N j WGrW.N /j !GrW.N /�j is an isomorphism.

If ` is an integer then.W.N /Œ`�/j DW.N /jC`.

THEOREM 66. Let ' W �� ! � nD be the period map of a variation of pureHodge structure of weightk with unipotent monodromyT D eN . Then, the limitHodge filtrationF1 of ' pairs with theweight monodromy filtrationM.N / WD

W.N /Œ�k� to define a mixed Hodge structure relative to whichN is a.�1;�1/-morphism.

REMARK 67. The limit Hodge filtrationF1 depends upon the choice of localcoordinates, or more precisely on the value of.ds/0. Therefore, unless one hasa preferred coordinate system (say, if the field of definitionmatters), in order


to extract geometric information from the limit mixed HodgestructureH1 D

.F1;M.N // one usually has to pass to the mixed Hodge structure induced byH1 on the kernel or cokernel ofN . In particular, ifX ! � is a semistabledegeneration, the local invariant cycle theorem asserts that we have an exactsequence

H k.X0/!H1

N!H1;

where the mapH k.X0/!H1 is obtained by first including the reference fiberXso

into X and then retractingX ontoX0.

The proof of Theorem 66 depends upon Schmid’s SL2-orbit theorem. Infor-mally, this result asserts that any 1-parameter nilpotent orbit is asymptotic toa nilpotent orbit arising from a representation of SL2.R/. In order to properlystate Schmid’s results we need to discuss splittings of mixed Hodge structures.

THEOREM68 (DELIGNE [De1]). Let.F;W / be a mixed Hodge structure onV .There exists a unique, functorial bigrading

VC DM

p;q

Ip;q

such that

(a)Fp DL

a�p Ia;b ;

(b) Wk DL

aCb�k Ia;b;(c) Ip;q D Iq;p mod

L

r<q;s<p I r;s .

In particular, if.F;W / is a mixed Hodge structure onV then.F;W / inducesa mixed Hodge structure ongl.V /Š V ˝V � with bigrading

gl.VC/DM

r;s

gl.V /r;s

wheregl.V /r;s is the subspace ofgl.V / which mapsIp;q to IpCr;qCs for all.p; q/. In the case where.F;W / is graded-polarized, we have an analogousdecompositiongC D

L

r;s gr;s of the Lie algebra ofGC.D Aut.VC;Q//. Forfuture use, we define

��1;�1.F;W /

DM

r;s<0

gl.V /r;s (4-4)

and note that by properties (a)–(c) of Theorem 68

� 2��1;�1.F;W /

) Ip;q

.e�:F;W /D e�:I

p;q

.F;W /: (4-5)

A mixed Hodge structure.F;W / is split overR if NIp;q D Iq;p for .p; q/. Ingeneral, a mixed Hodge structure.F;W / is not split overR. However, by atheorem of Deligne [CKS1], there is a functorial splitting operation

.F;W /‘ . OFı;W /D .e�iı:F;W /


which assigns to any mixed Hodge structure.F;W / a split mixed Hodge struc-ture. OFı;W /, such that

(a) ı D Nı,(b) ı 2��1;�1

.F;W /, and

(c) ı commutes with all.r; r/-morphisms of.F;W /.

REMARK 69.��1;�1.F;W /

D��1;�1

. OFı;W /.

A nilpotent orbit O�.z/ D ezN :F is an SL2-orbit if there exists a group homo-morphism� W SL2.R/! GR such that

O�.g:p�1/D �.g/: O�.

p�1/

for all g 2SL2.R/. The representation� is equivalent to the data of ansl2-triple.N;H;N C/ of elements inGR such that

ŒH;N �D�2N; ŒN C;N �DH; ŒH;N C�D 2N C

We also note that, for nilpotent orbits of pure Hodge structure, the statement thatezN :F is an SL2-orbit is equivalent to the statement that the limit mixed Hodgestructure.F;M.N // is split overR [CKS1].

THEOREM 70 (SL2-ORBIT THEOREM, [Sc]). Let�.z/D ezN :F be a nilpotentorbit of pure Hodge structure. Then, there exists a unique SL2-orbit O�.z/ DezN: OF and a distinguished real-analytic function

g.y/ W .a;1/!GR

(for somea 2 R) such that(a)�.iy/D g.y/: O�.iy/ for y > a, and(b) bothg.y/ andg�1.y/ have convergent series expansions about1 of the

form

g.y/D 1CX

k>0

gj y�k ; g�1.y/D 1CX

k>0

fky�k

with gk , fk 2 ker.adN /kC1.Furthermore, the coefficientsgk andfk can be expressed in terms of univer-

sal Lie polynomials in the Hodge components ofı with respect to. OF ;M.N //

andadN C.

REMARK 71. The precise meaning of the statement thatg.y/ is a distinguishedreal-analytic function, is thatg.y/ arises in a specific way from the solution ofa system of differential equations attached to� .


REMARK 72. If � is a nilpotent orbit of pure Hodge structures of weightk

and O� D ezN : OF is the associated SL2-orbit then. OF ;M.N // is split overR.The map.F;M.N //‘ . OF ;M.N // is called thesl2-splitting of .F;M.N //.Furthermore, OF D e�� :F where� is given by universal Lie polynomials in theHodge components ofı. In this way, one obtains ansl2-splitting .F;W / ‘

. OF ;W / for any mixed Hodge structure.F;W /.

4.3. Nilpotent and SL2-orbit theorems in the mixed case. In analogy tothe theory of period domains for pure HS, one can form a classifying spaceof graded-polarized mixed Hodge structureM with fixed Hodge numbers. Itspoints are the decreasing filtrationsF of the reference fiberV which pair withthe weight filtrationW to define a graded-polarized mixed Hodge structure (withthe given Hodge numbers). Given a variation of mixed Hodge structureV ofthis type over a complex manifoldS , one obtains a period map

� W S ! � nM:

M is a complex manifold upon which the Lie groupG, consisting of elements ofGL.VC/ which preserveW and act by real isometries on GrW , acts transitively.Next, let GC denote the Lie group consisting of elements of GL.VC/ whichpreserveW and act bycomplexisometries on GrW . Then, in analogy with thepure case, the “compact dual”LM of M is the complex manifold

LMŠGC=GFo

C

for any base pointFo 2M. The subgroupGR D G \GL.VR/ acts transitivelyon the real-analytic submanifoldMR consisting of pointsF 2M such that.F;W / is split overR.

EXAMPLE 73. LetM be the classifying space of mixed Hodge structures withHodge numbersh1;1 D h0;0 D 1. Then,MŠ C.

The proof of Schmid’s nilpotent orbit theorem depends critically upon the factthat the classifying spaceD has negative holomorphic sectional curvature alonghorizontal directions [GS]. Thus, although one can formally carry out all of theconstructions leading up to the statement of the nilpotent orbit theorem in themixed case, in light of the previous example it follows that one can not havenegative holomorphic sectional curvature in the mixed case, and hence thereis no reason to expect an analog of Schmid’s Nilpotent Orbit Theorem in themixed case. Indeed, for this classifying spaceM, the period map'.s/D exp.s/gives an example of a period map with trivial monodromy whichhas an essentialsingularity at1. Some additional condition is clearly required, and this iswhereadmissibility comes in.


In the geometric case of a degeneration of pure Hodge structure, Steenbrink[St] gave an alternative construction of the limit Hodge filtration that can beextended to variations of mixed Hodge structure of geometric origin [SZ]. Moregenerally, given anadmissiblevariation of mixed Hodge structureV over asmooth complex algebraic varietyS � NS such thatD D NS � S is a normalcrossing divisor, and any pointp 2D about whichV has unipotent local mon-odromy, one has an associated nilpotent orbit.e

P

j zj Nj :F1;W / with limitmixed Hodge structure.F1;M / whereM is the relative weight filtrationofN D

P

j Nj andW .11 Furthermore, one has the following “group theoretic”version of the nilpotent orbit theorem: As in the pure case, avariation of mixedHodge structureV ! .��/r with unipotent monodromy gives a holomorphicmap

W .��/r ! LM;

z ‘ e�P

zj Nj F.z/;

and this extends to�r if V is admissible. Let

q1 DM

r<0

gr;s

wheregC D Lie.GC/ DL

r;s gr;s relative to the limit mixed Hodge structure.F1;M /. Thenq1 is a nilpotent Lie subalgebra ofgC which is a vector spacecomplement to the isotropy algebragF1

C of F1. Consequently, there exists anopen neighborhoodU of zero ingC such that

U ! LM;

u‘ eu:F1

is a biholomorphism, and hence after shrinking�r as necessary we can write

.s/D e� .s/:F1

relative to a uniqueq1-valued holomorphic function� on�r which vanishesat 0. Recalling the construction of from the lifted period mapF , it followsthat

F.z1; : : : ; zr /D eP

j zj Nj e� .s/:F1:

This is called thelocal normal formof V atp and will be used in the calculationsof ~ 5.4–5.

There is also a version of Schmid’s SL2-orbit theorem for admissible nilpo-tent orbits. In the case of 1-variable and weight filtrationsof short length, the isdue to the second author in [Pe2]. More generally, Kato, Nakayama and Usui

11Recall [SZ] that in general the relative weight filtrationM D M.N;W / is the unique filtration (if itexists) such thatN.Mk/ � Mk�2 andM induces the monodromy weight filtration ofN on each GrW

i

(centered abouti).


proved a several variable SL2-orbit theorem with arbitrary weight filtration in[KNU1]. Despite the greater generality of [KNU1], in this paper we are goingto stick with the version of the SL2-orbit theorem from [Pe2] as it is sufficientfor our needs and has the advantage that for normal functions, mutatis mutandis,it is identical to Schmid’s result.

4.4. Outline of proof of Theorem 61. Let us now specialize to the case of anadmissible normal function� W S ! J.H/ over a curve and outline the proof[BP1] of Theorem 61. Before proceeding, we do need to addressone aspect ofthe SL2-orbit theorem in the mixed case. LetO� D .ezN :F;W / be an admissiblenilpotent orbit with limit mixed Hodge structure.F;M / which is split overR.Then, O� induces an SL2-orbit on each GrWk , and hence a correspondingsl2-representation�k .

DEFINITION 74. Let W be an increasing filtration, indexed byZ, of a finitedimensional vector spaceV . A grading of W is a direct sum decompositionWk D Vk ˚Wk�1 for each indexk.

In particular, a mixed Hodge structure.F;W / on V gives a grading ofW bythe ruleVk D

L

pCqDk Ip;q. Furthermore, if the ground field has characteristiczero, a grading ofW is the same thing as a semisimple endomorphismY of V

which acts as multiplication byk on Vk . If .F;W / is a mixed Hodge structurewe letY.F;W / denote the grading ofW which acts onIp;q as multiplication bypC q, theDeligne gradingof .F;W /.

Returning to the admissible nilpotent orbitO� considered above, we now havea system of representations�k on GrWk . To construct ansl2-representation onthe reference fiberV , we need to pick a gradingY of W . Clearly for each Hodgeflag F.z/ in the orbit we have the Deligne gradingY.F.z/;W /; but we are aftersomething more canonical. Now we also have the Deligne grading Y. OF ;M / ofM associated to thesl2-splitting of the LMHS. In the unpublished letter [De3],Deligne observed that:

THEOREM 75. There exists a unique gradingY of W which commutes withY. OF ;M / and has the property that if.N0;H;N

C

0/ denote the liftings of thesl2-

triples attached to the graded representations�k via Y thenŒN �N0;NC

0�D 0.

With this choice ofsl2-triple, andO� an admissible nilpotent orbit in 1-variable ofthe type arising from an admissible normal function, the main theorem of [Pe2]asserts that one has a direct analog of Schmid’s SL2-orbit theorem as statedabove for O� .

REMARK 76. More generally, given an admissible nilpotent orbit.ezN F;W /

with relative weight filtrationM DM.N;W /, Deligne shows that there exists


a gradingY D Y .N;Y.F;M // with similar properties. See [BP1] for details andfurther references.

REMARK 77. In the case of a normal function, if we decomposeN according toadY we haveN DN0CN�1 whereN�1 must be either zero or a highest weightvector of weight�1 for the representation ofsl2.R/ defined by.N0;H;N

C

0/.

Accordingly, since there are no vectors of highest weight�1, we haveN DN0

and henceŒY;N �D 0.

The next thing that we need to recall is that if� W S ! J.H/ is an admissiblenormal function which is represented by an extension

0!H ! V! Z.0/! 0

in the category of admissible variations of mixed Hodge structure onS then thezero locusZ of � is exactly the set of points where the corresponding DelignegradingY.F ;W/ is integral. In the case whereS � NS is a curve, in order toprove the algebraicity ofZ, all we need to do is show thatZ cannot contain asequence of pointss.m/ which accumulate to a puncturep 2 NS �S unless� isidentically zero. The first step towards the proof of Theorem61 is the followingresult [BP1]:

THEOREM78. Let' W��!� nM denote the period map of an admissible nor-mal function� W��! J.H/ with unipotent monodromy, andY be the gradingof W attached to the nilpotent orbit� of ' by Deligne’s construction(Theorem75). Let F W U !M denote the lifting of' to the upper half-plane. Then, forRe.z/ restricted to an interval of finite length, we have

limIm.z/!1

Y.F.z/;W / D Y

SKETCH OF PROOF. Using [De3], one can prove this result in the case where'

is a nilpotent orbit with limit mixed Hodge structure which is split overR. Letz D xC iy. In general, one writes

F.z/D ezN e� .s/:F1 D exN eiyN e� .s/e�iyN eiyN:F1

where exN is real, eiyN:F1 can be approximated by an SL2-orbit and thefunctioneiyN e� .s/e�iyN decays to1 very rapidly. ˜

In particular, if there exists a sequences.m/ which converges top along whichY.F ;W/ is integral it then follows from the previous theorem thatY is integral.An explicit computation then shows that the equation of the zero locus nearpis given by the equation

Ad.e� .s//Y D Y;

which is clearly holomorphic on a neighborhood ofp in NS .


That concludes the proof forS a curve. In the case whereS has a compact-ification NS such that NS �S is a smooth divisor, one can prove Theorem 61 bythe same techniques by studying the dependence of the preceding constructionson holomorphic parameters, i.e., at a point inD we get a nilpotent orbit

�.zI s2; : : : ; sr /D ezN:F1.s2; : : : ; sr /;

whereF1.s2; : : : ; sr / depend holomorphically on the parameters.s2; : : : ; sr /.

4.5. Zero loci and filtrations on Chow groups. Returning now to the al-gebraicity of the Hodge locus discussed at the beginning of this section, theHodge Conjecture would further imply that iff WX!S can be defined over analgebraically closed subfield ofC then so can the germ ofT . Claire Voisin [Vo1]gave sufficient conditions forT to be defined overNQ if f W X ! S is definedoverQ. Very recently F. Charles [Ch] carried out an analogous investigation ofthe field of definition of the zero locusZ of a normal function motivated overF. We reprise this last notion (from Sections 1 and 2):

DEFINITION 79. Let S be a smooth quasiprojective variety defined over asubfieldF0 � C, and letF � C be a finitely generated extension ofF0. Anadmissible normal function� 2 ANF.S;H/ is motivated overF if there existsa smooth quasiprojective varietyX , a smooth projective morphismf WX ! S ,and an algebraic cycleZ 2 Zm.X /prim, all defined overF, such thatH is asubVHS of.R2m�1f�Z/˝OS and� D �Z.

REMARK 80. HereZm.X /prim denotes algebraic cycles with homologicallytrivial restriction to fibers. One traditionally also assumesZ is flat overS , butthis can always be achieved by restricting toU � S sufficiently small (Zariskiopen); and then by [S1](i) �ZU

is NS admissible.Next, for anys0 2 S one canmoveZ by a rational equivalence to intersectXs0

(hence thefXsg for s in ananalytic neighborhood ofs0) properly, and then use the remarks at the beginningof [Ki] or [GGK, ~ III.B] to see that(ii) �Z is defined and holomorphic over allof S . Putting (i) and(ii) together with [BFNP, Lemma 7.1], we see that�Z isitself admissible.

Recall that the level of a VHSH is (for a generic fiberHs) the maximum dif-ferencejp1 � p2j for H p1;q1 andH p2;q2 both nonzero. A fundamental openquestion about motivic normal functions is then:

CONJECTURE81. (i) [ZL.D;E/] For everyF � C finitely generated overNQ,S=F smooth quasiprojective of dimensionD, andH! S VHS of weight.�1/

and level� 2E�1, the following holds: � motivated overF implies thatZ.�/ isan at most countable union of subvarieties ofS defined over(possibly different)finite extensions ofF.


(ii) [ fZL.D;E/] Under the same hypotheses, Z.�/ is an algebraic subvarietyof S defined over an algebraic extension ofF.

Clearly Theorem 57 and ConjectureZL.D;E/ together implyfZL.D;E/, butit is much more natural to phrase some statements (especially Proposition 86below) in terms ofZL.D;E/. If true even forDD1 (but generalE), Conjecture81(i) would resolve a longstanding question on the structure of Chow groups ofcomplex projective varieties. To wit, the issue is whether the second Bloch–Beilinson filtrand and the kernel of theAJ map must agree; we now wish todescribe this connection. We shall writefZL.D; 1/alg for the case when� ismotivated by a family of cycles algebraically equivalent tozero.

Let X be smooth projective andm2N. Denoting “ Q” by a subscriptQ, wehave the two “classical” invariantsclX ;Q WCH m.X /Q!Hgm.X /Q andAJX ;Q W

ker.clX ;Q/! J m.X /Q. It is perfectly natural both to ask for further Hodge-theoretic invariants for cycle-classes in ker.AJX ;Q/, and inquire as to what sortof filtration might arise from their successive kernels. Theidea of a (conjec-tural)systemof decreasing filtrations on the rational Chow groups ofall smoothprojective varieties overC, compatible with the intersection product, morphismsinduced by correspondences, and the algebraic Kunneth components of the diag-onal�X , was introduced by A. Beilinson [Be], and independently by S. Bloch.(One has to assume something like the Hard Lefschetz Conjecture so that theseKunneth components exist; the compatibility roughly says that Gri CH m.X /Qis controlled byH 2m�i.X /.) Such a filtrationF �

BB is unique if it exists and isuniversally known as aBloch–Beilinson filtration(BBF); there is a wide varietyof constructions which yield a BBF under the assumption of various more-or-less standard conjectures. The one which is key for the filtration (due to Lewis[Le2]) we shall consider is thearithmetic Bloch–Beilinson Conjecture(BBC):

CONJECTURE82. If X= NQ is a quasiprojective variety, the absolute-Hodgecycle-class map

cH W CH m.X /Q!H 2mH .X an

C ;Q.m// (4-6)

is injective. (HereCH m.X / denotes�rat-classes of cycles overNQ, and differsfrom CH m.XC/.)

Now for X=C, cH on CH m.X /Q is far from injective (the kernel usually noteven being parametrizable by an algebraic variety); but anygiven cycleZ 2

Zm.X / (a priori defined overC) is in fact defined over a subfieldK�C finitelygenerated overNQ, say of transcendence degreet . ConsideringX;Z overK, theNQ-spread then provides

� a smooth projective varietyNS= NQ of dimensiont , with NQ. NS/Š

! K ands0 W

Spec.K/! NS the corresponding generic point;


� a smooth projective varietyNX and projective morphismN� W NX ! NS , bothdefined overNQ, such thatX DXs0

WD X �s0Spec.K/; and

� an algebraic cycleNZ 2Zm.X. NQ// with Z D NZ�s0Spec.K/.

Writing N�smDW � W X ! S (andZ WD NZ\X ), we denote byU � S any affineZariski open subvariety defined overNQ, and putXU WD�

�1.U /, ZU WD NZ\XU ;note thats0 factors through all suchU .

The point is that exchanging the field of definition for additional geometryallowscH to “see” more; in fact, since we are overNQ, it should now (by BBC)see everything. NowcH.ZU / packages cycle-class and Abel–Jacobi invariantstogether, and the idea behind Lewis’s filtration (and filtrations of M. Saito andGreen/Griffiths) is to split the whole package up into Leray graded pieces withrespect to�. Miraculously, the0-th such piece turns out to agree with thefundamental class ofZ, and the next piece is the normal function generated byZU . The pieces after that define the so-calledhighercycle-class andAJ maps.

More precisely, we haveCH m.X.K //Q

spreadŠ››

WD

˘˘

imfCH m. NX /Q! lim�!U

CH m.XU /Qg

cH

››

H 2mHW im

�

H 2mD. NX an

C ;Q.m//! lim�!U

H 2mH..XU /

anC ;Q.m//

�

(4-7)

with cH (hence ) conjecturally injective. Lewis [Le2] defines a Leray filtrationL�H 2m

Hwith graded pieces

0

››

J 0�

lim�!U

W�1H i�1.U;R2m�i��Q.m//�

im lim�!U

Hg0�

GrW0 H i.U;R2m�i��Q.m//�

ˇ

››

GriL

H 2mH

˛

››

Hg0�

lim�!U

W0H i.U;R2m�i��Q.m//�

››

0

(4-8)


and setsLiCH m.XK /Q WD �1.LiH 2mH/. For Z 2 LiCH m.XK /Q, we put

cl iX.Z/ WD ˛.GriL .Z//; if this vanishes then Gri

L .Z/DWˇ.aj i�1X

.Z//, andvanishing ofcl i.Z/ andaj i�1.Z/ implies membership inLiC1. One easilyfinds thatcl0

X.Z/ identifies withclX ;Q.Z/ 2 Hg0.X /Q.

REMARK 83. The arguments of Hg0 andJ 0 in (4-8) have canonical and func-torial MHS by [Ar]. One should think of the top term as Gri�1

L of the lowest-weight part ofJ m.XU / and the bottom as Gri

L of the lowest-weight part ofHgm.XU / (both in the limit overU ).

Now to get a candidate BBF, Lewis takes

LiCH m.XC/Q WD lim

��!K�C

f:g:= NQ

LiCH m.XK /Q:

Some consequences of the definition of a BBF mentioned above,specificallythe compatibility with the Kunneth components of�X , include these:

(a)

8

ˆ

<

ˆ

:

F0BBCH m.X /Q D CH m.X /Q;

F1BBCH m.X /Q D CH m

hom.X /Q;

F2BBCH m.X /Q � ker.AJX ;Q/;

(b) FmC1BB CH m.X /D f0g.

These are sometimes stated as additional requirements for aBBF.

THEOREM 84 [Le2]. L� is intersection- and correspondence-compatible, andsatisfies(a). Assuming BBC, L� satisfies(b); and additionally assuming HLC,L� is a BBF.

The limits in (4-8) insideJ 0 and Hg0 stabilize for sufficiently smallU ; replacingS by such aU , we may consider the normal function�Z 2 ANF.S;H2m�1

X=S/

attached to theNQ-spread ofZ.

PROPOSITION85. (i) For i D 1, (4-8)becomes

0! J mfix.X=S/Q!Gr1L H 2m

H !�

H 1.S;R2m�1��Q/�.0;0/

! 0:

(ii) For Z 2 CH mhom.XK /Q, we havecl1

X.Z/ D Œ�Z�Q. If this vanishes, then

aj 0X.Z/DAJX .Z/Q 2 J m

fix.X=S/Q � J m.X /Q (implying L2 � kerAJQ).

So for Z 2 CH mhom.XK / with NQ-spreadZ over S , the information contained

in Gr1L.Z/ is (up to torsion) precisely�Z. Working overC, Z � Xs0

D Z

is the fiber of the spread at avery general points0 2 S.C/: trdeg. NQ.s0/= NQ/

is maximal, i.e., equal to the dimension ofS . SinceAJ is a transcendental(rather than algebraic) invariant, there is no outright guarantee that vanishing of


AJX .Z/ 2 J m.X /— or equivalently, of the normal function at a very generalpoint — implies the identical vanishing of�Z or evenŒ�Z�. To display explicitlythe depth of the question:

PROPOSITION86. (i) ZL.1;E/ .8E 2 N/ ” L2CH m.X /Q D ker.AJX ;Q/

.8 sm. proj. X=C/.(ii) ZL.1; 1/alg”L2CH m.X /Q\CH m

alg.X /QDker.AJX ;Q/\CH malg.X /Q

.8 sm. proj. X=C/.

Roughly speaking, these statements say that “sensitivity of the zero locus (of acycle-generated normal function) to field of definition” is equivalent to “spreadsof homologically andAJ -trivial cycles give trivial normal functions”. In (ii),the cycles in both statements are assumed algebraically equivalent to zero.

PROOF. We first remark that for any varietyS with field of definitionF of mini-mal transcendence degree, no properNF-subvariety ofS contains (in its complexpoints) a very general point ofS .

(i) .) / W Let Z be the NQ-spread ofZ with AJ.Z/Q D 0, and supposeGr1

L.Z/D Gr1

LcH.Z/ does not vanish. Taking a1-dimensional very general

multiple hyperplane sectionS0�S throughs0 (S0 is “minimally” defined over

ktrdeg:1� K), the restriction Gr1

LcH.Z0/ ¤ 0 by weak Lefschetz. Since each

Z.�N Z0/�S0 is a union of subvarieties defined overNk and containss0 for some

N 2N, one of these is all ofS0 (which implies Gr1L.Z/D 0), a contradiction.

SoZ 2 L2.

.(/ W Let X0! S0, Z0 2 Zm.X0/prim, dim.S0/ D 1, all be defined overkand supposeZ.�Z0

/ contains a points0 not defined overNk. Spreading this outover NQ to Z;X ;S �S0 3 s0, we have:s0 2S is very general,Z is the NQ-spreadof Z D Z0 �Xs0

, andAJ.Z/Q D 0. SoZ 2 L2 implies �Z is torsion, whichimplies �Z0

is torsion. But then�Z0is zero since it is zero somewhere (ats0).

SoZ.�Z0/ is eitherS0 or a (necessarily countable) union ofNk-points ofS0.

(ii) The spreadZ of Z.s0/ �alg 0 has every fiberZs �alg 0, hence�Z is asection ofJ.H/, H � .R2m�1��Q.m//˝OS a subVHS of level one (whichcan be taken to satisfyHs D .H 2m�1.Xs//h for a.e.s 2 S). The rest is asin (i). ˜

REMARK 87. A related candidate BBF which occurs in work of the first authorwith J. Lewis [KL, ~ 4], is defined via successive kernels ofgeneralizednormalfunctions (associated to theNQ-spreadZ of a cycle). These take values on verygeneral.i � 1/-dimensional subvarieties ofS (rather than at points), and havethe abovecl i.Z/ as their topological invariants.


4.6. Field of definition of the zero locus.We shall begin by showing that theequivalent conditions of Proposition 86(ii) are satisfied;the idea of the argumentis due in part to S. Saito [Sa]. The first paragraph in the following in fact givesmore:

THEOREM 88. fZL.D; 1/alg holds for allD 2 N. That is, the zero locus of anynormal function motivated by a family of cycles overF algebraically equivalentto zero, is defined over an algebraic extension ofF.

Consequently, cycles algebraically and Abel–Jacobi-equivalent to zero on asmooth projective variety overC, lie in the second Lewis filtrand.

PROOF. ConsiderZ 2 Zm.X /prim andf W X ! S defined overK (K beingfinitely generated overNQ), with Zs �alg 0 8s 2 S ; and lets0 2 Z.�Z/. (Note:s0 is just a complex point ofS .) We need to show:

9N 2 N such that�.s0/ 2 Z.�N Z/ for any� 2Gal.C=K/: (4-9)

Here is why (4-9) will suffice: the analytic closure of the setof all conjugatepoints is simply the point’sK-spreadS0 � S , a (possibly reducible) algebraicsubvariety defined overK. Clearly, on thes0-connected component ofS0, �Z

itself then vanishes; and this component is defined over an algebraic extension ofK. Trivially, Z.�Z/ is the union of such connected spreads of its pointss0; andsinceK is finitely generated overNQ, there are only countably many subvarietiesof S0 defined overK or algebraic extensions thereof. This provesZL.D; 1/alg,hence (by Theorem 57)fZL.D; 1/alg.

To show (4-9), writeX D Xs0, Z D Zs0

, andL.=K/ for their field of defi-nition. There exist overL

� a smooth projective curveC and points0; q 2 C.L/;� an algebraic cycleW 2Zm.C �X / such thatZ DW�.q� 0/; and� another cycle� 2Z1.J.C /�C / defining Jacobi inversion.Writing � WDW ı� 2Zm.J.C /�X /, the induced map

Œ�� W J.C /! J m.X /alg�

� J m.X /h�

is necessarily a morphism of abelian varieties overL; hence the identity con-nected component of ker.Œ��/ is a subabelian variety ofJ.C / defined over analgebraic extensionL0 � L. Define� WD�jB 2Zm.B �X /, and observe thatŒ� �� WB! J m.X /alg is zero by construction, so thatcl.�/ 2 L2H 2m.B �X /.

Now, sinceAJX .Z/D 0, a multipleb WD N:AJC .q � 0/ belongs toB, andthenN:Z D ��b. This “algebraizes” theAJ -triviality of N:Z: conjugating the6-tuple.s0;Z;X;B; �; b/ to .�.s0/;Z

� ŒDZ�.s0/�;X� ŒDX�.s0/�;B

� ; �� ; b� /;

we still haveN:Z� D �� b� andcl.�� / 2 L2H 2m.B� �X � / by motivicity ofthe Leray filtration [Ar], and this impliesN:AJ.Z� /D Œ�� b� D 0 as desired.

˜


We now turn to the result of [Ch] indicated at the outset of~ 4.5. While inter-esting, it sheds no light onZL.1;E/ or filtrations, since the hypothesis that theVHS H have no global sections is untenable over a point.

THEOREM89 [Ch, Theorem 3].LetZ be the zero locus of ak-motivated normalfunction� W S ! J.H/. Assume thatZ is algebraic andHC has no nonzeroglobal sections overZ. ThenZ is defined over a finite extension ofk.

PROOF. Charles’s proof of this result uses the`-adic Abel–Jacobi map. Al-ternatively, we can proceed as follows (using, withF D k, the notation ofDefinition 79): takeZ0 � Z.�/ to be an irreducible component (without lossof generality assumed smooth), andZZ0

the restriction ofZ to Z0. Let ŒZZ0�

and ŒZZ0�dR denote the Betti and de Rham fundamental classes ofZZ0

, andL the Leray filtration. Then, Gr1

LŒZZ0

� is the topological invariant ofŒZZ0� in

H 1.Z0;R2m�1f�Z/, whereas Gr1

LŒZZ0

�dR is the infinitesimal invariant of�Z

overZ0. In particular, sinceZ0 is contained in the zero locus of�Z,

GrjLŒZZ0

�dR D 0; j D 0; 1: (4-10)

Furthermore, by the algebraicity of the Gauss–Manin connection, (4-10) is in-variant under conjugation:

GrjLŒZZ

�

0�dR D .Grj

LŒZZ0

�dR/�

and hence GrjLŒZZ

�

0�dR D 0 for j D 0, 1. Therefore, Grj

LŒZZ

�

0� D 0 for j D

0, 1, and henceAJ.Zs/ takes values in the fixed part ofJ.H/ for s 2 Z�0

.By assumption,HC has no fixed part overZ0, and hence no fixed part overZ�

0(since conjugation mapsr-flat sections tor-flat sections by virtue of the

algebraicity of the Gauss–Manin connection). As such, conjugation must takeus to another component ofZ, and hence (sinceZ is algebraic overC impliesZhas only finitely many components),Z0 must be defined over a finite extensionof k. ˜

We conclude with a more direct analog of Voisin’s result [Vo1, Theorem 0.5(2)]on the algebraicity of the Hodge locus. IfV is a variation of mixed Hodgestructure over a complex manifold and

˛ 2 .Fp \W2p \VQ/so

for someso 2 S , then the Hodge locusT of ˛ is the set of points inS wheresome parallel translate of belongs toFp.

REMARK 90. If .F;W / is a mixed Hodge structure onV andv2Fp\W2p\VQ

thenv is of type.p;p/ with respect to Deligne’s bigrading of.F;W /.


THEOREM 91. Let S be a smooth complex algebraic variety defined over asubfieldk of C, andV be an admissible variation of mixed Hodge structure ofgeometric origin overS . Suppose thatT is an irreducible subvariety ofS overC such that:

(a)T is an irreducible component of the Hodge locus of some

˛ 2 .Fp \W2p \VQ/toI

(b) �1.T; to/ fixes only the line generated by.

Then, T is defined overNk.

PROOF. If V Š Q.p/ for somep then T D S . Otherwise,T cannot be anisolated point without violating (b). Assume therefore that dimT > 0. OverT , we can extend to a flat family of de Rham classes. By the algebraicity ofthe Gauss–Manin connection, the conjugate˛� is flat overT � . Furthermore, ifT � supports any additional flat families of de Rham classes, conjugation by��1

gives a contradiction to (b). Therefore,˛� D�ˇ, whereˇ is a�1.T� /-invariant

Betti class onT � which is unique up to scaling. Moreover,

Q.˛; ˛/DQ.˛� ; ˛� /D �2Q.ˇ; ˇ/

and hence there are countably many Hodge classes that one canconjugate tovia Gal.C=k/. Accordingly,T must be defined overNk. ˜

5. The Neron model and obstructions to singularities

The unifying theme of the previous sections is the study of algebraic cyclesvia degenerations using the Abel–Jacobi map. In particular, in the case of asemistable degeneration� W X ! � and acohomologically trivialcycle Z

which properly intersects the fibers, we have

lims!0

AJXs.Zs/DAJX0

.Z0/

as explained in detail in~ 2. In general however, the existence of the limit Abel–Jacobi map is obstructed by the existence of the singularities of the associatednormal function. Nonetheless, using the description of theasymptotic behaviorprovided by the nilpotent and SL2-orbit theorems, we can define the limits ofadmissible normal functions along curves and prove the algebraicity of the zerolocus.

5.1. Neron models in one parameter.In this section we consider the problemof geometrizing these constructions (ANFs and their singularities, limits andzeroes) by constructing a Neron model which graphs admissible normal func-tions. The quest to construct such objects has a long historywhich traces back tothe work of Neron on minimal models for abelian varietiesAK defined over the


field of fractionsK of a discrete valuation ringR. In [Na], Nakamura proved theexistence of an analytic Neron model for a family of abelian varietiesA!��

arising from a variation of Hodge structureH! �� of level 1 with unipotentmonodromy. With various restrictions, this work was then extended to normalfunctions arising from higher codimension cycles in the work of Clemens [Cl2],El Zein and Zucker [EZ], and Saito [S1].

REMARK 92. Unless otherwise noted, throughout this section we assume thatthe local monodromy of the variation of Hodge structureH under considerationis unipotent, and the local systemHZ is torsion free.

A common feature in all of these analytic constructions of Neron models forvariations of Hodge structure over�� is that the fiber over02� is a complex Liegroup which has only finitely many components. Furthermore,the componentinto which a given normal function� extends is determined by the value of�Z;0.�/. Using the methods of the previous section, one way to see this is asfollows: Let

0!H! V! Z.0/! 0

represent an admissible normal function� W��!J.H/ andF WU !M denotethe lifting of the period map ofV to the upper half-plane, with monodromyT D eN . Then, using the SL2-orbit theorem of the previous section, it follows(cf. Theorem 4.15 of [Pe2]) that

YHodgeD limIm.z/!1

e�zN :Y.F.z/;W /

exists, and is equal to the gradingY .N;Y.F1;M // constructed in the previoussection; recall also thatY .N;Y.F1;M // 2 ker.adN / due to the short lengthof the weight filtration. Suppose further that there exists an integral gradingYBetti 2 ker.adN / of the weight filtrationW . Let j W��!� andi W f0g !�

denote the inclusion maps. Then,YHodge�YBetti defines an element in

J.H0/D Ext1MHS.Z.0/;H0.i�Rj�

H// (5-1)

by simply applyingYHodge�YBetti to any lift of 1 2Z.0/DGrW0 . Reviewing~ 2and~ 3, we see that the obstruction to the existence of such a grading YBetti isexactly the class�Z;0.�/.

REMARK 93. More generally, ifH is a variation of Hodge structure of weight�1 over a smooth complex algebraic varietyS and NS is a good compactificationof S , given a points 2 NS we define

J.Hs/D Ext1MHS.Z;Hs/; (5-2)


whereHs DH 0.i�s Rj�H/ andj W S! NS , is W fsg! NS are the inclusion maps.

In case NSnS is a NCD in a neighborhood ofS , with fNig the logarithms of theunipotent parts of the local monodromies, thenHs Š

T

j ker.Nj /.

In general, except in the classical case of degenerations ofHodge structure oflevel 1, the dimension ofJ.H0/ is usually strictly less than the dimension ofthe fibers ofJ.H/ over��. Therefore, any generalized Neron modelJ�.H/ ofJ.H/ which graphs admissible normal functions cannot be a complex analyticspace. Rather, in the terminology of Kato and Usui [KU; GGK],we obtain a“slit analytic fiber space”. In the case where the base is a curve, the observationsabove can be combined into the following result:

THEOREM94. LetH be a variation of pure Hodge structure of weight�1 over asmooth algebraic curveS with smooth projective completionNS . Let j W S ! NSdenote the inclusion map. Then, there exists a Neron model forJ.H/, i.e., atopological groupJ NS .H/ over NS satisfying the following two conditions:

(i) J NS .H/ restricts toJ.H/ overS .(ii) There is a one-to-one correspondence between the set of admissible normal

functions� W S ! J.H/ and the set of continuous sectionsN� W NS ! J NS .H/

which restrict to holomorphic, horizontal sections ofJ.H/ overS .

Furthermore:

(iii) There is a short exact sequence of topological groups

0! J NS .H/0! J NS .H/! G! 0;

whereGs is the torsion subgroup of.R1j�HZ/s for anys 2 NS .

(iv) J NS .H/0 is a slit analytic fiber space, with fiberJ.Hs/ overs 2 NS .

(v) If � W S ! J.H/ is an admissible normal function with extensionN� then theimage ofN� in Gs at the points 2 NS �S is equal to�Z;s.�/. Furthermore, if�Z;s.�/D 0 then the value ofN� at s is given by the class ofYHodge�YBetti asin (5-1). Equivalently, in the geometric setting, if �Z;s.�/D 0 then the valueof N� at s is given by the limit Abel–Jacobi map.

Regarding the topology of the Neron model, let us consider more generally thecase of smooth complex varietyS with good compactificationNS , and recall from~ 2 that we have also have the Zucker extensionJ Z

NS.H/ obtained by starting from

the short exact sequence of sheaves

0!HZ!HO=F0! J.H/! 0

and replacingHZ by j�HZ andHO=F0 by its canonical extension. Following

[S5], let us suppose thatD D NS � S is a smooth divisor, and letJ ZNS.H/Inv

Dbe

the subset ofJ ZNS.H/ defined by the local monodromy invariants.


THEOREM95 [S5].The Zucker extensionJ ZNS.H/ has the structure of a complex

Lie group over NS , and it is a Hausdorff topological space on a neighborhood ofJ Z

NS.H/Inv

D.

Specializing this result to the case whereS is a curve, we then recover the resultof the first author together with Griffiths and Green thatJ NS .H/

0 is Hausdorff,since in this case we can identifyJ NS .H/

0 with J ZNS.H/Inv

D.

REMARK 96. Using this Hausdorff property, Saito was able to prove in[S5] thealgebraicity of the zero locus of an admissible normal function in this setting(i.e., D smooth).

5.2. Neron models in many parameters.To extend this construction further,we have to come to terms with the fact that unlessS has a compactificationNS such thatD D NS � S is a smooth divisor, the normal functions that weconsider may have nontorsion singularities along the boundary divisor. Thiswill be reflected in the fact that the fibersGs of G need no longer be finitegroups. The first test case is whenH is a Hodge structure of level 1. In thiscase, a Neron model forJ.H/ was constructed in the thesis of Andrew Young[Yo]. More generally, in joint work with Patrick Brosnan andMorihiko Saito,the second author proved the following result:

THEOREM 97 [BPS].Let S be a smooth complex algebraic variety andH bea variation of Hodge structure of weight�1 overS . Let j W S ! NS be a goodcompactification ofNS andfS˛g be a Whitney stratification ofNS such that

(a)S is one of the strata ofNS , and(b) theRkj�HZ are locally constant on each stratum.

Then, there exists a generalized Neron model forJ.H/, i.e., a topologicalgroupJ NS .H/ over NS which extendsJ.H/ and satisfies these two conditions:

(i) The restriction ofJ NS .H/ to S is J.H/.(ii) Any admissible normal function� W S ! J.H/ has a unique extension to a

continuous sectionN� of J NS .H/.

Furthermore:

(iii) There is a short exact sequence of topological groups

0! J NS .H/0! J NS .H/! G! 0

over NS such thatGs is a discrete subgroup of.R1j�HZ/s for any points 2 NS .(iv) The restriction ofJ NS .H/

0 to any stratumS˛ is a complex Lie group overS˛ with fiberJ.Hs/ overs 2 NS .

(v) If � W S ! J.H/ is an admissible normal function with extensionN� then theimage ofN�.s/ in Gs is equal to�Z;s.�/ for all s 2 NS . If �Z;s.�/ D 0 for alls 2 NS then N� restricts to a holomorphic section ofJ NS .H/

0 over each strata.


REMARK 98. More generally, this is true under the following hypothesis:(1) S is a complex manifold andj W S ! NS is a partial compactification of

S as an analytic space;(2) H is a variation of Hodge structure onS of negative weight, which need

not have unipotent monodromy.

To construct the identity componentJ NS .H/0, let� WS!J.H/ be an admissible

normal function which is represented by an extension

0!H! V! Z.0/! 0 (5-3)

andj W S ! NS denote the inclusion map. Also, givens 2 NS let is W fsg ! NS

denote the inclusion map. Then, the short exact sequence (5-3) induces an exactsequence of mixed Hodge structures

0!Hs!H 0.i�

s Rj�V/! Z.0/!H 1.i�

s Rj�H/; (5-4)

where the arrowZ.0/!H 1.i�s Rj�H/ is given by1‘ �Z;s.�/. Accordingly,

if �Z;s.�/D 0 then (5-4) determines a pointN�.s/ 2 J.Hs/. Therefore, as a set,we define

J NS .H/0 D

a

s2 NS

J.Hs/

and topologize by identifying it with a subspace of the Zucker extensionJ ZNS.H/.

Now, by condition (b) of Theorem 97 and the theory of mixed Hodge mod-ules[S4], it follows that ifi˛ WS˛! NS are the inclusion maps thenH k.i�

˛Rj�H/

are admissible variations of mixed Hodge structure over each stratumS˛. Inparticular, the restriction ofJ NS .H/

0 to S˛ is a complex Lie group.Suppose now that� W S ! J.H/ is an admissible normal function with ex-

tensionN� W NS ! J NS .H/ such that�Z;s.�/D 0 for eachs 2 NS . Then, in order toprove thatN� is a continuous section ofJ NS .H/

0 which restricts to a holomorphicsection over each stratum, it is sufficient to prove thatN� coincides with thesection of the Zucker extension (cf. [S1, Proposition 2.3]). For this, it is in turnsufficient to consider the curve case by restriction to the diagonal curve�!�r

by t ‘ .t; : : : ; t/; see [BPS,~ 1.4].It remains now to constructJ NS .H/ via the following gluing procedure: Let

U be an open subset ofNS and� WU ! J.H/ be an admissible normal functionwith cohomological invariant

�Z;U .�/D @.1/ 2H 1.U;HZ/

defined by the map

@ WH 0.U;Z.0//!H 1.U;HZ/


induced by the short exact sequence (5-3) overU. Then, we declareJU .HU \S /�

to be the component ofJ NS .H/ overU , and equipJU .HU \S /� with a canonical

morphismJU .HU \S /

�! JU .HU \S /0

which sends� to the zero section. If� is another admissible normal functionoverU with �Z;U .�/D �Z;U .�/ then there is a canonical isomorphism

JU .HU \S /� Š JU .HU \S /

�

which corresponds to the section� �� of JU .HU \S /0 overU .

Addendum to ~ 5.2. Since the submission of this article, there have been severalimportant developments in the theory of Neron models for admissible normalfunctions on which we would like to report here. To this end, let us supposethatH is a variation of Hodge structure of level 1 over a smooth curve S � NS .Let AS denote the corresponding abelian scheme with Neron modelA NS overNS . Then, we have a canonical morphism

A NS ! J NS .H/

which is an isomorphism overS . However, unlessH has unipotent local mon-odromy about each points 2 NS�S , this morphism is not an isomorphism [BPS].Recently however, building upon his work on local duality and mixed Hodgemodules [Sl2], Christian Schnell has found an alternative construction of theidentity component of a Neron model which contains the construction of [BPS]in the case of unipotent local monodromy and agrees [SS] withthe classicalNeron model for VHS of level 1 in the case of nonunipotent monodromy. Inthe paragraphs below, we reproduce a summary of this construction which hasbeen generously provided by Schnell for inclusion in this article.

The genesis of the new construction is in unpublished work ofClemens onnormal functions associated to primitive Hodge classes. When Y is a smoothhyperplane section of a smooth projective varietyX of dimension2n, andHZ D H 2n�1.Y;Z/van its vanishing cohomology modulo torsion, the interme-diate JacobianJ.Y / can be embedded into a bigger object,K.Y / in Clemens’snotation, defined as

K.Y /D

�

H 0�

X; ˝2nX.nY /

�_

H 2n�1.Y;Z/van:

The point is that the vanishing cohomology ofY is generated by residues ofmeromorphic2n-forms onX , with the Hodge filtration determined by the orderof the pole (provided thatOX .Y / is sufficiently ample). Clemens introducedK.Y / with the hope of obtaining a weak, topological form of Jacobiinversionfor its points, and because of the observation that the numerator in its definition


makes senseeven whenY becomes singular. In his Ph.D. thesis [Sl3], Schnellproved that residues and the pole order filtration actually give a filtered holo-nomicD-module on the projective space parametrizing hyperplane sections ofX ; and that thisD-module underlies the polarized Hodge module correspondingto the vanishing cohomology by Saito’s theory. At least in the geometric case,therefore, there is a close connection between the questionof extending inter-mediate Jacobians, and filteredD-modules (with the residue calculus providingthe link).

The basic idea behind Schnell’s construction is to generalize from the geo-metric setting above to arbitrary bundles of intermediate Jacobians. As before,let H be a variation of polarized Hodge structure of weight�1 on a complexmanifoldS , andM its extension to a polarized Hodge module onNS . Let.M;F /

be its underlying filtered leftD-module:M is a regular holonomicD-module,andF D F�M a good filtration by coherent subsheaves. In particular,F0M

is a coherent sheaf onNS that naturally extends the Hodge bundleF0HO. Nowconsider the analytic space overNS , given by

T D T .F0M/D SpecNS

�

SymO NS.F0M/

�

;

whose sheaf of sections is.F0M/_. (OverS , it is nothing but the vector bundlecorresponding to.F0HO/

_.) It naturally contains a copyTZ of theetale space ofthe sheafj�HZ; indeed, every point of that space corresponds to a local sectionof HZ, and it can be shown that every such section defines a map ofD-modulesM!O NS via the polarization.

Schnell proves thatTZ � T is a closed analytic subset, discrete on fibers ofT ! NS . This makes the fiberwise quotient spaceNJ D T=TZ into an analyticspace, naturally extending the bundle of intermediate Jacobians forH . He alsoshows that admissible normal functions with no singularities extend uniquely toholomorphic sections ofNJ ! NS . To motivate the extension process, note thatthe intermediate Jacobian of a polarized Hodge structure ofweight�1 has twomodels,

HC

F0HCCHZ

'.F0HC/

_

HZ

;

with the isomorphism coming from the polarization. An extension of mixedHodge structure of the form

0!H ! V ! Z.0/! 0 (5-5)

gives a point in the second model in the following manner.Let H �DHom.H;Z.0// be the dual Hodge structure, isomorphic toH.�1/

via the polarization. After dualizing, we have

0! Z.0/! V �!H �! 0;


and thus an isomorphismF1V �

C ' F1H �

C ' F0HC. Therefore, anyv 2 VZ

lifting 12Z gives a linear mapF0HC!C, well-defined up to elements ofHZ;this is the point in the second model ofJ.H / that corresponds to the extensionin (5-5).

It so happens that this second construction is the one that extends to all of NS .Given a normal function� on S , let

0!HZ! VZ! ZS ! 0

be the corresponding extension of local systems. By applying j�, it gives anexact sequence

0! j�HZ! j�VZ! Z NS !R1j�HZ;

and when� has no singularities, an extension of sheaves

0! j�HZ! j�VZ! Z NS ! 0:

Using duality for filteredD-modules, one obtains local sections of.F0M/_

from local sections ofj�VZ, just as above, and thus a well-defined holomorphicsection of NJ ! NS that extends�.

As in the one-variable case, where the observation is due to Green, Griffiths,and Kerr, horizontality constrains such extended normal functions to a certainsubset of NJ ; Schnell proves that this subset is precisely the identity componentof the Neron model constructed by Brosnan, Pearlstein, and Saito. With theinduced topology, the latter is therefore a Hausdorff space, as expected. Thisprovides an additional proof for the algebraicity of the zero locus of an admissi-ble normal function, similar in spirit to the one-variable result in Saito’s paper,in the case when the normal function has no singularities.

The other advance, is the recent construction [KNU2] of log intermediateJacobians by Kato, Nakayama and Usui. Although a proper exposition of thistopic would take us deep into logarithmic Hodge theory [KU],the basic idea isas follows: LetH! �� be a variation of Hodge structure of weight�1 withunipotent monodromy. Then, we have a commutative diagram

J.H/Q'

//

››

Q� nM

GrW�1

››

��'

// � nD

(5-6)

where Q' and' are the respective period maps. In [KU], Kato and Usui explainedhow to translate the bottom row of this diagram into logarithmic Hodge theory.More generally, building on the ideas of [KU] and the severalvariable SL2-orbittheorem [KNU1], Kato, Nakayama and Usui are able to construct a theory of


logarithmic mixed Hodge structures which they can then apply to the top rowof the previous diagram. In this way, they obtain a log intermediate Jacobianwhich serves the role of a Neron model and allows them to give an alternateproof of Theorem 57 [KNU3].

5.3. Singularities of normal functions overlying nilpotent orbits. We nowconsider the group of componentsGs of J NS .H/ at s 2 NS . For simplicity, wefirst consider the case whereH is a nilpotent orbitHnilp over .��/r . To thisend, we recall that in the case of a variation of Hodge structure H over .��/r

with unipotent monodromy, the intersection cohomology ofHQ is computed bythe cohomology of a complex.B�.N1; : : : ;Nr /; d/ (cf. ~ 3.4). Furthermore, theshort exact sequence of sheaves

0!HQ! VQ!Q.0/! 0

associated to an admissible normal function� W .��/r ! J.H/ with unipotentmonodromy gives a connecting homomorphism

@ W IH0.Q.0//! IH1.HQ/

such [email protected]/D Œ.N1.e

Qo /; : : : ;Nr .e

Qo /�D sing0.�/;

whereeQo is an element in the reference fiberVQ of VQ overso 2 .�

�/r whichmaps to1 2 Q.0/. After passage to complex coefficients, the admissibility ofV allows us to pick an alternate lifteo 2 VC to be of type.0; 0/ with respectto the limit MHS ofV . It also forceshj D Nj .eo/ to equalNj .fj / for someelementfj 2HC of type.0; 0/ with respect to the limit MHS ofH. Moreover,e

Q

0� e0 DW h maps to0 2GrW0 , hence lies inHC, so

.N1.eQ

0/; : : : ;Nr .e

Q

0//� .N1.e0/; : : : ;Nr .e0// modulod.B0/D im

rL

jD1

Nj

(i.e., up to.N1.h/; : : : ;Nr .h//).

COROLLARY 99. sing0.�/ is a rational class of type.0; 0/ in IH1.HQ/.

SKETCH OF PROOF. This follows from the previous paragraph together withthe explicit description of the mixed Hodge structure on thecohomology ofB�.N1; : : : ;Nr / given in [CKS2]. ˜

Conversely, we have:

LEMMA 100. LetHnilp D eP

j zj Nj:F1 be a nilpotent orbit of weight�1 over��r with rational structureHQ. Then, any class of type.0; 0/ in IH1.HQ/ isrepresentable by aQ-normal function� which is an extension ofQ.0/ byHnilp

such thatsing0.�/D ˇ.


PROOF. By the remarks above, corresponds to a collection of elementshj 2

Nj .HC/ such that

(a) h1; : : : ; hr are of type.�1;�1/ with respect to the limit mixed Hodgestructure ofHnilp,

(b) d.h1; : : : ; hr /D 0, i.e.,Nj .hk/�Nk.hj /D 0, and(c) There existsh 2HC such thatNj .h/Chj 2HQ for eachj , i.e., the class

of .h1; : : : ; hr / in IH1.HC/ belongs to the imageIH1.HQ/! IH1.HC/.

We now define the desired nilpotent orbit by formally settingVC D Ceo ˚

HC, whereeo is of type.0; 0/ with respect to the limit mixed Hodge structureand lettingVQ D Q.eoC h/˚HQ. We defineNj .eo/ D hj . Then, followingKashiwara [Ka]:

(a) The resulting nilpotent orbitVnilp is pre-admissible.(b) The relative weight filtration of

W�2 D 0; W�1 DHQ; W0 D VQ

with respect to eachNj exists.

ConsequentlyVnilp is admissible, and the associated normal function� hassingularityˇ at 0. ˜

5.4. Obstructions to the existence of normal functions withprescribed sin-gularity class. Thus, in the case of a nilpotent orbit, we have a complete de-scription of the group of components of the Neron model Q. In analogy withnilpotent orbits, one might expect that given a variation ofHodge structureH ofweight�1 over.��/r with unipotent monodromy, the group of components ofthe Neron model Q to equal the classes of type.0; 0/ in IH1.HQ/. However,Saito [S6] has managed to construct examples of variations of Hodge structureover.��/r which do not admit any admissible normal functions with nontorsionsingularities. We now want to describe Saito’s class of examples. We beginwith a discussion of the deformations of an admissible nilpotent orbit into anadmissible variation of mixed Hodge structure over.��/r .

Let ' W .��/r ! � nD be the period map of a variation of pure Hodge struc-ture with unipotent monodromy. Then, after lifting the period map ofH to theproduct of upper half-planesU r , the work of Cattani, Kaplan and Schmid ondegenerations of Hodge structure gives us a local normal form of the period map

F.z1; : : : ; zr /D eP

j zj Nj e� .s/:F1:

Here,.s1; : : : ; sr / are the coordinates on�r , .z1; : : : ; zr / are the coordinates onU r relative to which the covering mapU r ! .��/r is given bysj D e2� izj ;


� W�r ! gC

is a holomorphic function which vanishes at the origin and takes values in thesubalgebra

qDM

p<0

gp;qI

andL

p;q gp;q denotes the bigrading of the MHS induced ongC (cf. ~ 4.2) bythe limit MHS .F1;W .N1 C � � �Nr /Œ1�/ of H. The subalgebraq is gradednilpotent

qDM

a<0

qa; qa DM

b

ga;b;

with N1; : : : ;Nr 2 q�1. Therefore,

eP

j zj Nj e� .s/ D eX .z1;:::;zr /;

whereX takes values inq, and hence the horizontality of the period map be-comes

e�X @eX D @X�1;

whereX D X�1 CX�2 C � � � relative to the grading ofq. Equality of mixedpartial derivatives then forces

@X�1 ^ @X�1 D 0

Equivalently,�

Nj C 2� isj@��1

@sj; Nk C 2� isk

@��1

@sk

�

D 0: (5-7)

REMARK 101. The function� and the local normal form of the period mapappear in [CK].

In his letter to Morrison [De4], Deligne showed that for VHS over .��/r withmaximal unipotent boundary points, one could reconstruct the VHS from dataequivalent to the nilpotent orbit and the function��1. More generally, one canreconstruct the function� starting from@X�1 using the equation

@eX D eX @X�1

subject to the integrability condition@X�1^@X�1D0. This is shown by Cattaniand Javier Fernandez in [CF].

The above analysis applies to VMHS over.��/r as well: As discussed inthe previous section, a VMHS is given by a period map from the parameterspace into the quotient of an appropriate classifying spaceof graded-polarizedmixed Hodge structureM. As in the pure case, we have a Lie groupG whichacts onM by biholomorphisms and a complex Lie groupGC which acts on the“compact dual” LM.


As in the pure case (and also discussed in~ 4), an admissible VMHS withnilpotent orbit.e

P

j zj Nj :F1;W / will have a local normal form

F.z1; : : : ; zr /D eP

j zj Nj e� .s/:F1;

where� W�r ! gC takes values in the subalgebra

qDM

p<0

gp;q:

Conversely (given an admissible nilpotent orbit), subjectto the integrabilitycondition (5-7) above, any function��1 determines a corresponding admissibleVMHS; see [Pe1, Theorem 6.16].

Returning to Saito’s examples (which for simplicity we onlyconsider in thetwo-dimensional case), letH be a variation of Hodge structure of weight�1

over�� with local normal formF.z/ D ezN e� .s/:F1. Let � W �2 ! � by�.s1; s2/D s1s2. Then, for��.H/, we have

��1.s1; s2/D ��1.s1s2/:

In order to construct a normal function, we need to extend��1.s1; s2/ andN1DN2DN on the reference fiberHC of H to include a new classu0 of type.0; 0/ which projects to1 in Z.0/. Set

N1.u0/D h1; N2.u0/D h2; ��1.s1; s2/u0 D ˛.s1; s2/:

Note that.h1; h2/ determines the cohomology class of the normal function soconstructed, and thath2 � h1 depends only on the cohomology class, and notthe particular choice of representative.h1; h2/.

In order to construct a normal function in this way, we need tocheck hori-zontality. This amounts to checking the equation

N

�

s2

@˛

@s2

� s1

@˛

@s1

�

C s1s2�0

�1.s1s2/.h2�h1/

C 2� is1s2�0

�1.s1s2/

�

s2

@˛

@s2

� s1

@˛

@s1

�

D 0:

Computation shows that the coefficient of.s1s2/m on the left side is

1

.m� 1/!�.m/

�1.0/.h2�h1/: (5-8)

Hence, a necessary condition for the cohomology class represented by.h1; h2/

to arise from an admissible normal function is forh2�h1 to belong to the kernelof ��1.t/. This condition is also sufficient since, under this hypothesis, one cansimply set D 0.


EXAMPLE 102. LetX�! � be a family of Calabi–Yau3-folds (smooth over

��, smooth total space) with Hodge numbersh3;0 D h2;1 D h1;2 D h0;3 D 1

and central singular fiber having an ODP. SettingH WDH3X�=��.2/, the LMHS

has as its nonzeroIp;q ’s I�2;1; I�1;�1; I0;0; and I1;�2. Assume that theYukawa coupling.rıs

/3 2 HomO�.H

3;0e ;H

0;3e / is nonzero (ıs D s d=ds), and

thus the restriction of��1.s/ to HomO�.I�1;�1; I�2;1/, does not vanish iden-

tically. Then, for any putative singularity class,0 ¤ h2 � h1 2 .I�1:�1/Q Š

ker.N /.�1;�1/Q (this being isomorphic to (2-10) in this case, which is just one-

dimensional) for admissible normal functions overlying��H, nonvanishing of��1.s/.h2�h1/ on� implies that (5-8) cannot be zero for everym.

5.5. Implications for the Griffiths–Green conjecture. Returning now to thework of Griffiths and Green on the Hodge conjecture via singularities of normalfunctions, it follows using the work of Richard Thomas that for a sufficientlyhigh power ofL, the Hodge conjecture implies that one can force�� to have asingularity at a pointp 2 OX such that��1.p/ has only ODP singularities. Ingeneral, on a neighborhood of such a pointOX need not be a normal crossingdivisor. However, the image of the monodromy representation is neverthelessabelian. Using a result of Steenbrink and Nemethi [NS], it then follows from theproperties of the monodromy cone of a nilpotent orbit of pureHodge structurethat singp.��/ persists under blowup. Therefore, it is sufficient to study ODPdegenerations in the normal crossing case (cf. [BFNP, sec. 7]). What we willfind below is that the “infinitely many” conditions above (vanishing of (5-8) forall m) are replaced by surjectivity of a single logarithmic Kodaira–Spencer mapat each boundary component. Consequently, as suggested in the introduction, itappears that M. Saito’s examples are not a complete show-stopper for existenceof singularities for Griffiths–Green normal functions.

The resulting limit mixed Hodge structure is of the form

I0;0

� � � I�2;1 I�1;0 I0;�1 I1;�2 � � �

I�1;�1

andN 2 D 0 for every element of the monodromy coneC. The weight filtrationis given by

M�2.N /DX

j

Nj .HC/; M�1.N /D\

j

ker.Nj /; M0.N /DHC:

For simplicity of notation, let us restrict to a two parameter version of such adegeneration, and consider the obstruction to constructing an admissible normalfunction with cohomology class represented by.h1; h2/ as above. As in Saito’s


example, we need to add a classuo of type .0; 0/ such thatNj .uo/ D hj andconstruct˛ D ��1.uo/. Then, the integrability condition@X�1 ^ @X�1 D 0

becomes

�.2� is2/@��1

@s2

.h1/C .2� is1/@��1

@s1

.h2/

C .2� is1/.2� is2/

�

@��1

@s1

@˛

@s2

�@��1

@s2

@˛

@s1

�

D 0; (5-9)

since˛ D ��1.uo/ takes values inM�1.N /.Write ˛ D

P

j ;k sj1sk

2 jk and��1 DP

p;q sp1

sq2 pq on HC. Then, forab

nonzero, the coefficient ofsa1sb

2on the left side of equation (5-9) is

�2� ib ab.h2/C 2� ia ab.h1/C .2� i/2X

pCjDaqCkDb

.pk � qj / pq. jk/:

Define

�ab D 2� ib ab.h2/� 2� ia ab.h1/� .2� i/2X

pCjDaqCkDbpq¤0

.pk � qj / pq. jk/:

Then, equation (5-9) is equivalent to

.2� i/2b 10.˛.a�1/b/� .2� i/2a 01.˛a.b�1//D �ab;

where jk occurs in�ab only in total degreej C k < aC b � 1. Therefore,providedthat

10; 01 W F�11 =F0

1! F�21 =F�1

1

are surjective, we can always solve (nonuniquely!) for the coefficients jk ,and hence formally (i.e., modulo checking convergence of the resulting series)construct the required admissible normal function with given cohomology class.

REMARK 103. (i) Of course, it is not necessary to have only ODP singularitiesfor the analysis above to apply. It is sufficient merely that the limit mixed Hodgestructure have the stated form. In particular, this is always true for degenerationsof level 1. Furthermore, in this case Gr�2

F1D 0, and hence, after tensoring with

Q, the group of components of the Neron model surjects onto the Tate classesof type.0; 0/ in IH1.HQ/.

(ii) In Saito’s examples from~ 5.4, even if� 0

�1.0/¤ 0, we will have 01 D

0 D 10, since the condition of being a pullback via.s1; s2/ ‘ s1s2 means��1.s1; s2/D

P

p;q sp1

sq2 pq D

P

r sr1sr

2 rr .


EXAMPLE 104. In the case of a degeneration of Calabi–Yau threefolds withlimit mixed Hodge structure on the middle cohomology (shifted to weight�1)

I0;0

I�2;1 I�1;0 I0;�1 I1;�2

I�1;�1

the surjectivity of the partial derivatives of��1 are related to the Yukawa cou-pling as follows: Let

F.z/D eP

j zj Nj e� .s/:F1

be the local normal form of the period map as above. Then, a global nonvan-ishing holomorphic section of the canonical extension ofF1 (i.e., ofF3 beforewe shift to weight�1) is of the form

˝ D eP

j zj Nj e� .s/�1.s/;

where�1 W �r ! I1;�2 is holomorphic and nonvanishing. Then, the Yukawa

coupling of˝ is given by

Q.˝;Dj DkD`˝/; Da D@

@za:

In keeping with the notation above, leteX D eP

j zj Nj e� .s/ andAj DDj X�1.Using the first Hodge–Riemann bilinear relation and the factthateX is an auto-morphism ofQ, it follows that

Q.˝;Dj DkD`˝/DQ.�1.s/;Aj AkA` �1.s//:

Moreover (cf. [CK; Pe1]), the horizontality of the period map implies that�

��1jsk D0 ;Nk

�

D 0

Using this relation, it then follows that

lims!0

Q.˝;Dj DkD`˝/

.2� isj /.2� isk/.2� is`/DQ.�1.0/;Gj GkG`�1.0//

for j ¤ k, where

Ga D@��1

@sa.0/:

In particular, if for each indexj there exist indicesk and` with k ¤ ` such thatthe left-hand side of the previous equation is nonzero thenGj W .F

�11=F0

1/!

.F�21=F�1

1/ is surjective.


6. Global considerations: monodromy of normal functions

Returning to a normal functionV 2 NF1.S;H/adNS

over acompletebase, wewant to speculate a bit about how one might “force” singularities to exist. The(inconclusive) line of reasoning we shall pursue rests on two basic principles:

(i) maximality of the geometric (global) monodromy group ofV may bededuced from hypotheses on the torsion locus ofV ; and

(ii) singularities ofV can be interpreted in terms of the local monodromy ofV being sufficiently large.

While it is unclear what hypotheses (if any) would allow one to pass fromglobal to local monodromy-largeness, the proof of the first principle is itself ofinterest as a first application of algebraic groups (the algebraic variety analog ofLie groups, originally introduced by Picard) to normal functions.

6.1. Background. Mumford–Tate groups of Hodge structures were introducedby Mumford [Mu] for pure HS and by Andre [An] in the mixed setting. Theirpower and breadth of applicability is not well-known, so we will first attempta brief summary. They were first brought to bear onH 1.A/ for A an abelianvariety, which has led to spectacular results:

� Deligne’s theorem [De2] thatQ-Bettiness of a class inFpH2p

dR.Ak/ for k

algebraically closed is independent of the embedding ofk into C (“Hodgeimplies absolute Hodge”);

� the proofs by Hazama [Ha] and Murty [Mr] of the HC forA “nondegenerate”(MT of H 1.A/ is maximal in a sense to be defined below); and

� the density of special (Shimura) subvarieties in Shimura varieties and thepartial resolution of the Andre–Oort Conjecture by Klingler and Yafaev [KY].

More recently, MT groups have been studied for higher weightHS’s; one canstill use them to define specialNQ-subvarieties of (non-Hermitian-symmetric)period domainsD, which classify polarized HS’s with fixed Hodge numbers(and polarization). In particular, the0-dimensional subdomains — still dense inD — correspond to HS with CM (complex multiplication); that is, with abelianMT group. One understands these HS well: their irreducible subHS may beconstructed directly from primitive CM types (and have endomorphism algebraequal to the underlying CM field), which leads to a complete classification; andtheir Weil and Griffiths intermediate Jacobians are CM abelian varieties [Bo].Some further applications of MT groups include:

� Polarizable CM-HS are motivic [Ab]; when they come from a CY variety,the latter often has good modularity properties;


� Given H � of a smooth projective variety, the level of the MT Lie algebrafurnishes an obstruction to the variety being dominated by aproduct of curves[Sc];

� Transcendence degree of the space of periods of a VHS (over a baseS),viewed as a field extension ofC.S/ [An];

and specifically in the mixed case:

� the recent proof [AK] of a key case of the Beilinson–Hodge Conjecture forsemiabelian varieties and products of smooth curves.

The latter paper, together with [An] and [De2], are the best references for thedefinitions and properties we now summarize.

To this end, recall that an algebraic groupG over a fieldk is an algebraicvariety overk together withk-morphisms of varieties1G WSpec.k/!G, “mul-tiplication” �G W G �G ! G, and “inversion”{G W G ! G satisfying obviouscompatibility conditions. The latter ensure that for any extensionK=k, theK-pointsG.K/ form a group.

DEFINITION 105. (i) A (k-)closed algebraic subgroupM � G is one whoseunderlying variety is (k-)Zariski closed.

(ii) Given a subgroupM�G.K/, thek-closure ofM is the smallestk-closedalgebraic subgroupM of G with K-pointsM.K/ �M.

If M WDM.K/ for an algebraick-subgroupM � G, then thek-closure ofM is just thek-Zariski closure ofM (i.e., the algebraic variety closure).

But in general, this is not true: instead,M may be obtained as thek-Zariski(algebraic variety) closure of the group generated by thek-spread ofM.

We refer the reader to [Sp] (especially Chapter 6) for the definitions of reductive,semisimple, unipotent, etc. in this context (which are lesscrucial for the sequel).We will write DG WD ŒG;G� .E G/ for the derived group.

6.2. Mumford–Tate and Hodge groups. Let V be a (graded-polarizable)mixed Hodge structure with dualV _ and tensor spaces

T m;nV WD V ˝m˝ .V _/˝n

(n;m 2 Z�0). These carry natural MHS, and anyg 2 GL.V / acts naturally onT m;nV .

DEFINITION 106. (i) A Hodge.p;p/-tensoris any� 2 .T m;nV /.p;p/Q .

(ii) The MT groupMV of V is the (largest)Q-algebraic subgroup of GL.V /fixing12 the Hodge.0; 0/-tensors for allm; n. The weight filtrationW� on V ispreserved byMV .

12“Fixing” means fixing pointwise; the term for “fixing as a set”is “stabilizing”.


Similarly, theHodge groupM ı

Vof V is theQ-algebraic subgroup of GL.V /

fixing the Hodge.p;p/-tensors for allm; n;p. (In an unfortunate coincidence ofterminology, these are completely different objects from —though not unrelatedto — the finitely generated abelian groups Hgm.H / discussed in~ 1.)

(iii) The weight filtration onV induces one on MT/Hodge:

W�iM.ı/VWD˚

g 2M.ı/V

ˇ

ˇ .g� id/W�V �W��iV

E M.ı/V:

One has:W0M.ı/VDM

.ı/V

; W�1M.ı/V

is unipotent; and GrW0 M

.ı/VŠM

.ı/

V split

(V split WDL

`2Z GrW` V ), cf. [An].

ClearlyM ı

VE M

V; and unlessV is pure of weight0, we haveM

V=M ı

VŠGm.

If V has polarizationQ2HomMHS .V ˝V;Q.�k// for k 2Znf0g, thenM ı

Vis

of finite index inMV \GL.V;Q/ (whereg 2 GL.V;Q/ meansQ.gv;gw/DQ.v; w/), and if in additionV .D H / is pure (or at least split) then both arereductive. One has in general thatW�1MV �DMV �M ı

V�M

V.

DEFINITION 107. (i) If MV is abelian (” MV .C/Š .C�/�a), V is called a

CM-MHS. (A subMHS of a CM-MHS is obviously CM.)(ii) The endomorphisms EndMHS.V / can be interpreted as theQ-points of the

algebra.End.V //MV DWEV . One always hasMV �GL.V;EV / (Dcentralizerof EV ); if this is an equality, thenV is said to benondegenerate.

Neither notion implies the other; however: any CM or nondegenerate MHS is(Q-)split, i.e.,V .D V split/ is a direct sum of pure HS in different weights.

REMARK 108. (a) We point out why CM-MHS are split. IfMV is abelian,thenMV � EV and soMV .Q/ consists of morphisms of MHS. But then anyg 2W�1MV .Q/, henceg� id, is a morphism of MHS with.g� id/W��W��1;sog D id, andMV DMV split, which impliesV D V split.

(b) For an arbitrary MHSV , the subquotient tensor representations ofMV

killing DMV (i.e., factoring through the abelianization) are CM-MHS. By (a),they are split, so thatW�1MV acts trivially; this givesW�1MV �DMV .

Now we turn to the representation-theoretic point of view onMHS. Define thealgebraicQ-subgroupsU � S �GL2 via their complex points:

S.C/ W

��

˛ ˇ

�ˇ ˛

�ˇ

ˇ

ˇ

ˇ

˛; ˇ 2 C

.˛; ˇ/¤ .0; 0/

�

Š

eigenvalues// C� �C�

�

z;1

z

�

U.C/ W?

ffl

OO

��

˛ ˇ

�ˇ ˛

� ˇ

ˇ

ˇ

ˇ

˛; ˇ 2 C

˛2Cˇ2 D 1

�

Š//

C�?

ffl

OO

z_

OO

(6-1)


where the top map sends�

˛�ˇ

ˇ˛

�

‘ .˛C iˇ; ˛� iˇ/DW .z; w/. (Points inS.C/

will be represented by the “eigenvalues”.z; w/.) Let

' W S.C/!GL.VC/

be given by

'.z; w/jI p;q.H / WDmultiplication byzpwq .8p; q/:

Note that this map is in general only defined overC, though in the pure caseit is defined overR (and asS.R/ � S.C/ consists of tuples.z; Nz/, one tendsnot to see precisely the approach above in the literature). The following usefulresult13 allows one to compute MT groups in some cases.

PROPOSITION109.MV is theQ-closure of'.S.C// in GL.V /.

REMARK 110. In the pure (V D H ) case, this condition can be replaced byMH .R/ � '.S.R//, andM ı

Hdefined similarly as theQ-closure of'.U.R//;

unfortunately, forV a non-Q-split MHS theQ-closure of'.U.C// is smallerthanM ı

H.

Now let H be a pure polarizable HS with Hodge numbershp;q, and takeD

(with compact dualLD) to be the classifying space for such. We may viewLD asa quasiprojective variety overQ in a suitable flag variety. Consider the subgroupM ı

H ;'�M ı

Hwith real pointsM ı

H ;'.R/ WD .M ı

H.R//'.S.R//. If we view M ı

H

as acting on a Hodge flag ofHC with respect to a (fixed) basis ofHQ;thenM ı

H ;'is the stabilizer of the Hodge flag. This leads to a Noether–Lefschetz-

type substratum inD:

PROPOSITION111.The MT domain

DH WDM ı

H.R/

M ı

H ;'.R/

�

�M ı

H.C/

M ı

H ;'.C/DW LDH

�

classifies HS with Hodge group contained inMH , or equivalently with Hodge-tensor set containing that ofH . The action ofM ı

HuponH embedsLDH Œ LD

as a quasiprojective subvariety, defined over an algebraic extension ofQ. TheGL.HQ;Q/-translates of LDH give isomorphic subdomains(with conjugate MTgroups) dense in LD.

A similar definition works for certain kinds of MHS. The trouble with applyingthis in the variational setting (which is our main concern here), is that the “tauto-logical VHS” (or VMHS) over such domains (outside of a few classical cases inlow weight or level) violate Griffiths transversality henceare not actually VHS.

13Proof of this, and of Proposition 111 below, will appear in a work of the first author with P. Griffithsand M. Green.


Still, it can happen that MT domains in non-Hermitian symmetric period do-mains are themselves Hermitian symmetric. For instance, taking Sym3 of HS’sembeds the classifying space (Š H) of (polarized) weight-1 Hodge structureswith Hodge numbers.1; 1/ into that of weight-3 Hodge structures with Hodgenumbers.1; 1; 1; 1/.

6.3. MT groups in the variational setting. Let S be a smooth quasiprojec-tive variety with good compactificationNS , andV 2 VMHS.S/ad

NS; assumeV is

graded-polarized, which means we have

Q 2M

i

HomVMHS.S/�

.GrWi V/˝2;Q.�i/�

satisfying the usual positivity conditions. The Hodge flag embeds the universalcover OS.“ S/ in a flag variety; let theimage-pointof Os0.‘ s0/ be of maximaltranscendence degree. (One might says0 2 S.C/ is a “very general point in thesense of Hodge”; we arenot sayings0 is of maximal transcendence degree.)Parallel translation along the local systemV gives rise to the monodromy rep-resentation� W �1.S; s0/! GL.Vs0;Q;W�;Q/. Moreover, taking as basis forVs;Q the parallel translate of one forVs0;Q, MVs

is constant on paths (froms0/

avoiding a countable unionT of proper analytic subvarieties ofS , where in factSı WD SnT is pathwise connected. (At pointst 2 T , MVt

�MVs; and even the

MT group of the LMHS sV at x 2 NSnS naturally includes inMVs.)

DEFINITION 112. (i) We callMVs0DWMV the MT group, andM ı

Vs0DWM ı

V

theHodge group, of V . One has EndMHS.Vs0/Š EndVMHS.S/.V/; see [PS2].

(ii) The identity connected componentV of theQ-closure of�.�1.S; s0// isthe geometric monodromy group ofV ; it is invariant under finite coversQS “ S

(and semisimple in the split case).

PROPOSITION113. (Andre) ˘V E DMV .

SKETCH OF PROOF. By a theorem of Chevalley, any closedQ-algebraic sub-group of GL.Vs0

/ is the stabilizer, for some multitensort 2L

i T mi ;ni .Vs0;Q/

of Q hti. For MV , we can arrange for thistV to be itself fixed and to lie inL

i

�

T mi ;ni .Vs0/�.0;0/

Q. By genericity ofs0, Q htVi extends to a subVMHS with

(again by9 of Q) finite monodromy group, and sotV is fixed by˘V . Thisproves˘V �MV (in fact,�M ı

Vsince monodromy preservesQ). Normality

of this inclusion then follows from the Theorem of the Fixed Part: the largestconstant sublocal system of anyT m;n.V/ (stuff fixed by˘V ) is a subVMHS,hence subMHS ats0 and stable underMV .

Now let

M abV WD

MV

DMV

; ˘abV WD

˘V

˘V \DMV

�M abV ;


(which is a connected component of theQ-closure of some�ab�Mab;ıV

.Z/),and (taking a more exotic route than Andre) V ab be the (CM)MHS correspond-ing to a faithful representation ofM ab

V. For each irreducibleH � V ab, the

imageM abV

has integer pointsŠO�

Lfor some CM fieldL, andM

ab;ıV

.Q/� L

consists of elements of norm1 under any embedding. The latter generateL

(a well-known fact for CM fields) but, by a theorem of Kronecker, have finiteintersection withO�

L: the roots of unity. It easily follows from this that ab

V,

hence abV

, is trivial. ˜

DEFINITION 114. Letx 2 NS with neighborhood.��/k ��n�k in S and local(commuting) monodromy logarithmsfNig;14 define the weight monodromy fil-trationM x

�WDM.N;W /� whereN WD

PkiD1 Ni . In the following we assume

a choice of path froms0 to x:(a) Write�x

Vfor the local monodromy groupin GL.Vs0;Z;W�;Q/ generated

by theTi D .Ti/sseNi , and�x for the corresponding representation.(b) We say thatV is nonsingular atx if Vs0

ŠL

j GrWj Vs0as�x-modules. In

this case, the condition that sV ŠL

j s GrWj V is independent of the choiceof local coordinates.s1; : : : ; sn/ at x, andV is calledsemisplit (nonsingular)atx when this is satisfied.

(c) The GrMx

i sV are always independent ofs. We say thatV is totallydegenerate (TD)atx if these GrMi are (pure) Tate andstrongly degenerate(SD)atx if they are CM-HS. Note that the SD condition is interesting already for thenonboundary points (x 2 S; k D 0).

We can now generalize results of Andre [An] and Mustafin [Ms].

THEOREM115. If V is semisplit TD(resp. SD) at a pointx 2 NS , then˘V DM ı

V

(resp. DM ı

V).

REMARK 116. Note that semisplit SD atx 2 S simply means thatVx is a CM-MHS (this case is done in [An]). Also, if V DM ı

Vthen in fact V DDM ı

VD

M ı

V.

PROOF. Passing to a finite cover to identifyV and�.�1/, if we can show thatany invariant tensort 2

�

T m;nVs0;Q

�

˘V is also fixed byM ı

V(resp.DM ı

V), we

are done by Chevalley. Now the span ofM ı

Vt is (since V EM ı

V) fixed by�.�1/,

and (using the Theorem of the Fixed Part) extends to a constant subVMHSU � T m;nV DW T . Now the hypotheses onV carry over toT and taking LMHSatx, U D sU D

L

i s GrWi U DL

i GrWi U , we see thatU splits (as VMHS).As T is TD (resp. SD) atx, U is split Hodge–Tate (resp. CM-MHS).

14Though this has been suppressed so far throughout this paper, one hasfNi g and LMHS even in the gen-eral case where the local monodromiesTi are only quasi-unipotent, by writingTi DW .Ti /ss.Ti /u uniquelyas a product of semisimple and unipotent parts (Jordan decomposition) and settingNi WD log..Ti /u/.


If U is H-T then it consists of Hodge tensors; soM ı

Vacts trivially onU hence

on t.If U is CM thenM ı

VjU DM ı

Uis abelian; and so the action ofM ı

VonU factors

throughM ı

V=DM ı

V, so thatDM ı

Vfixes t. ˜

A reason why one would want this “maximality” resultVDM ı

Vis to satisfy the

hypothesis of the following interpretation of Theorem 91 (which was a partialgeneralization of results of [Vo1] and [Ch]). Recall that a VMHS V=S is k-motivated if there is a familyX ! S of quasiprojective varieties defined overk with Vs D the canonical (Deligne) MHS onH r .Xs/ for eachs 2 S .

PROPOSITION117. SupposeV is motivated overk with trivial fixed part, andlet T0 � S be a connected component of the locus whereM ı

Vsfixes some vector

(in Vs). If T0 is algebraic(overC), M ı

VT0hasonly one fixed line, and˘VT0

D

M ı

VT0, thenT0 is defined overNk.

Of course, to be able to use this one also needs a result on algebraicity of T0,i.e., a generalization of the theorems of [CDK] and [BP3] to arbitrary VMHS.Onenow has thisby work of Brosnan, Schnell, and the second author:

THEOREM 118. Given any integral, graded-polarizedV 2 VMHS.S/adNS, the

components of the Hodge locus of any˛ 2 Vs yield complex algebraic subvari-eties ofS .

6.4. MT groups of (higher) normal functions. We now specialize to the casewhereV 2 NFr .S;H/ad

NS, with H! S the underlying VHS of weight�r . M ı

V

is then an extension ofM ı

HŠM ı

Vsplit.DH˚QS .0//by (and a semidirect product

with) an additive (unipotent) group

U WDW�r M ı

V ŠG��a ;

with � � rankH. SinceM ı

Vrespects weights, there is a natural map� WM ı

V“

M ı

Hand one might ask when this is an isomorphism.

PROPOSITION119.�D 0” V is torsion.

PROOF. First we note thatV is torsion if and only if, for some finite coverQS “ S , we have

f0g ¤ HomVMHS. QS/.QS .0/;V/D EndVMHS. QS/

.V/\ann.H/

D EndMHS.Vs0/\ann.Hs0

/D�

HomQ..Vs0=Hs0

/;Vs0/�M ı

V :

The last expression can be interpreted as consisting of vectorsw 2 Hs0;Q thatsatisfy.id�M /w D u whenever

�1u

0M

�

2M ı

V. This is possible only if there

is oneu for eachM , i.e., if � W M ı

V! M ı

His an isomorphism. Conversely,


assuming this, writeuD ��1.M /�

noting

Q��1.M1M2/D Q��1.M1/CM1 Q�

�1.M2/�

.�/

and setw WD Q��1.0/. TakingM2D 0 andM1DM in .�/, we get.id�M /wD

.id�M /��1.0/D Q��1.M /D u for all M 2M ı

H. ˜

We can now address the problem which lies at the heart of this section: whatcan one say about the monodromy of the normal function above and beyond thatof the underlying VHS — for example, about the kernel of the natural map� W˘V “˘H? One can make some headway simply by translating Definition 114and Theorem 115 into the language of normal functions; all vanishing conditionsare˝Q.

PROPOSITION120.LetV be an admissible higher normal function overS , andlet x 2 NS with local coordinate systems.

(i) V is nonsingular(as AVMHS) at x if and only ifsingx.V/D 0. Assumingthis, V is semi-simple atx if and only iflimx.V/D0. (In casex2S , singx.V/D

0 is automatic andlimx.V/D 0 if and only ifx is in the torsion locus ofV .)(ii) V is TD(resp. SD) at x if and only if the underlying VHSH is. (For x 2S ,

this just means thatHx is CM.)(iii) If singx.V/, limx.V/ vanish and sH is graded CM, then˘V DDMV .

(For x 2 S , we are just hypothesizing that the torsion locus ofV contains a CMpoint ofH.)

(iv) Letx 2 NSnS . If singx.V/, limx.V/ vanish and sH is Hodge–Tate, then˘V DM ı

V.

(v) Under the hypotheses of(iii) and(iv), dim.ker.�//D �. (In general onehas�.)

PROOF. All parts are self-evident except for (v), which follows from observing(in both cases (iii) and (iv)) via the diagram

G��a ŠW�1M

.ı/VD ker.�/�DMV ˘V

ffl

�

//

�››

››

M.ı/V

�››

››

˘Hffl

�

// M.ı/H

(6-2)

that ker.�/D ker.�/. ˜

EXAMPLE 121. The Morrison–Walcher normal function from~ 1.7 (Exam-ple 13) lives “over” the VHSH arising fromR3��Z.2/ for a family of “mirrorquintic” CY 3-folds, and vanishes atzD1. (One should take a suitable, e.g., or-der 2 or 10 pullback so thatV is well-defined.) The underlying HSH at this pointis of CM type

�

the fiber is the usual.Z=5Z/3 quotient offP4

iD0 Z5i D 0g�P4

�

,


with MH .Q/ŠQ.�5/. SoV would satisfy the conditions of Proposition 120(iii).It should be interesting to work out the consequences of the resulting equality˘V DDMV .

There is a different aspect to the relationship between local and global behaviorof V . Assuming for simplicity that the local monodromies atx are unipotent, let�x WD ker.�x

V“ �x

H/ denote the local monodromy kernel, and�x the dimen-

sions of itsQ-closure�x. This is an additive (torsion-free) subgroup of ker.�/,and so dim.ker.�//��x (8x 2 NSnS). Writing fNig for the local monodromylogarithms atx, we have the

PROPOSITION122. (i) �x > 0 impliessingx.V/¤ 0 (nontorsion singularity)(ii ) The converse holds assumingr D 1 andrank.Ni/D 1 .8i/.

PROOF. Let g 2 �xV

, and definem 2 Q˚k by log.g/ DWPk

iD1 miNi . WritingNg, NNi for gjH, Ni jH , consider the (commuting) diagram of morphisms of MHS

sHL

NNi

wwo

o

o

o

o

o

o

o

o

o

o

o

„

r

$$

I

I

I

I

I

I

I

I

I

log. Ng/

rr

L

i sH.�1/

�’’

O

O

O

O

O

O

O

O

O

O

O

sV

L

Nioo

log.g/zzu

u

u

u

u

u

u

u

u

sH.�1/

(6-3)

where�.w1; : : : ; wk/DPk

iD1 miwi and log.g/DPk

iD1 miNNi . Then singx.V/

is nonzero if and only if�L

Ni

�

�Q does not lie in im.˚ NNi/, where�Q (seeDefinition 2(b)) generates sV= sH.

(i) Supposeg 2 �xnf1g. Then0 D log. Ng/ implies 0 D �.im.˚ NNi// while0¤ logg implies0¤ .log.g//�Q D �..˚Ni/�Q/: So� “detects” a singularity.

(ii) If r D 1 we may replaceLk

iD1 sH.�1/ in the diagram by the subspaceLk

iD1.Ni. sH//: Since each summand is of dimension 1, and�L

Ni

�

�Q … im�L

NNi

�

(by assumption), we can choosem D fmig in order that� kill im�L

NNi

�

butnot

�L

Ni

�

�Q. Using the diagram, log. Ng/D 0¤ log.g/ impliesg 2 �xnf1g. ˜

REMARK 123. (a) The existence of a singularityalwaysimplies thatV is non-torsion, hence� > 0.

(b) In the situation of [GG], we haver D 1 and rank 1 local monodromylogarithms; hence, by Proposition 122(ii), the existence of a singularity impliesdim.ker.�// > 0, consistent with (a).


(c) By Proposition 122(i), in the normal function case (r D 1), �x D 0 alongcodimension-1 boundary components.

(d) In the “maximal geometric monodromy” situation of Proposition 120(v),�� x 8x 2 NSnS .

Obviously, for the purpose of forcing singularities to exist, the inequality in (d)points in the wrong direction. One wonders if some sort of cone or spread on aVMHS might be used to translate global into local monodromy kernel, but thisseems unlikely to be helpful.

We conclude with an amusing application of differential Galois theory relatedto a result of Andre [An]:

PROPOSITION124.Consider a normal functionV of geometric origin togetherwith an OS -basisf!ig of holomorphic sections ofF0H. (That is, Vs is theextension of MHS corresponding toAJ.Zs/ 2 J p.Xs/ for some flat family ofcycles on a family of smooth projective varieties overS .) Let K denote theextension ofC.S/ by the(multivalued) periods of thef!ig; and L denote thefurther extension ofK via the(multivalued) Poincare normal functions given bypairing the!i with an integral lift of1 2 QS .0/ (i.e., the membrane integralsR

�s!i.s/ where@�s DZs). Thentrdeg.L=K/D dim.ker.�//.

The proof rests on a result of N. Katz [Ka, Corollary 2.3.1.1]relating tran-scendence degrees and dimensions of differential Galois groups, together withthe fact that thef

R

�s!ig (for eachi) satisfy a homogeneous linear ODE with

regular singular points [Gr1]. (This fact implies equalityof differential Galoisand geometric monodromy groups, since monodromy invariantsolutions of suchan ODE belong toC.S/which is the fixed field of the Galois group.) In the eventthatH has no fixed part (so thatL can introduce no new constants and one hasa “Picard–Vessiot field extension”) and the normal functionis motivated overk D Nk, one can probably replaceC by k in the statement.

References

[Ab] S. Abdulali, Hodge structures of CM type, J. Ramanujan Math. Soc. 20 (2005),no. 2, 155–162.

[An] Y. Andre, Mumford–Tate groups of mixed Hodge structures and the theorem ofthe fixed part, Compositio Math. 82 (1992), no. 1, 1–24.

[Ar] D. Arapura,The Leray spectral sequence is motivic, Invent. Math. 160 (2005), no.3, 567–589.

[AK] D. Arapura and M. Kumar,Beilinson–Hodge cycles on semiabelian varieties,Math. Res. Lett. 16 (2009), no. 4, 557–562.


[BBD] A. Beilinson, J. Bernstein, and P. Deligne,Faisceaux pervers, in “Analysis andtopology of singular spaces, I” (Luminy, 1981), pp. 5–171, Asterisque 100, Soc.Math. France, Paris, 1982.

[Be] A. A. Beilinson, Height pairing between algebraic cycles, in “K-theory, arith-metic, and geometry (Moscow, 1984–6)”, pp. 1–25, LNM 1289, Springer, Berlin,1987.

[Bl] S. Bloch, Algebraic cycles and higherK-theory, Adv. Math. 61 (1986), no. 3,267–304.

[Bo] C. Borcea,Calabi–Yau threefolds and complex multiplication, in “Essays onmirror manifolds”, 489–502, Intl. Press, Hong Kong, 1992.

[BP1] P. Brosnan and G. Pearlstein,The zero-locus of an admissible normal function,Ann. Math. 170 (2009), 883–897.

[BP2] ———, Zero loci of normal functions with torsion singularities, Duke Math. J.150 (2009), no. 1, 77–100.

[BP3] ———, On the algebraicity of the zero-locus of an admissible normal function,arXiv:0910.0628v1.

[BFNP] P. Brosnan, H. Fang, Z. Nie, and G. Pearlstein,Singularities of admissiblenormal functions, Inventiones Math. 117 (2009), 599–629.

[BPS] P. Brosnan, G. Pearlstein and M. Saito,A generalization of the Neron models ofGriffiths, Green and Kerr, arXiv:0809.5185, preprint.

[BPSc] P. Brosnan, G. Pearlstein and C. Schnell,The locus of Hodge classes in anadmissible variation of mixed Hodge structure, C. R. Acad. Sci. Paris, Ser. I 348(2010), 657–660.

[BZ] J.-L. Brylinski and S. Zucker,An overview of recent advances in Hodge theory,in “Several complex variables, VI”, 39–142, Encyclopedia Math. Sci. 69, Springer,Berlin, 1990.

[Ca] J. Carlson,Extensions of mixed Hodge structures, in “Journees de GeometrieAlgebrique d’Angers” (1979), pp. 107–127, Sijthoff & Noordhoff, Alphen aan defRijn—Germantown, Md., 1980.

[CDK] E. Cattani, P. Deligne, and A Kaplan,On the locus of Hodge classes, JAMS 8(1995), 483–506.

[CF] E. Cattani and J. Fernandez,Asymptotic Hodge theory and quantum products,math.AG/001113, in “Advances in algebraic geometry motivated my physics” (Low-ell, MA), pp. 115–136, Contemp. Math. 276, AMS, Providence,RI, 2001.

[CK] E. Cattani and A. Kaplan,Degenerating variations of Hodge structure, in “Actesdu Colloque de Theorie de Hodge” (Luminy), Asterisque 179–180 (1989), 9, 67–96.

[CKS1] E. Cattani, A. Kaplan, and W. Schmid,Degeneration of Hodge structures, Ann.of Math. 123 (1986), 457–535.

[CKS2] ———, L2 and intersection cohomologies for a polarizable variationofHodge structure, Invent. Math. 87 (1987), 217–252.


[CMP] J. Carlson, S. Muller-Stach, and C. Peters, “Period mappings and period do-mains”, Cambridge Stud. Adv. Math. 85, Cambridge University Press, Cambridge,2003.

[Ch] F. Charles,On the zero locus of normal functions and theetale Abel–Jacobi map,IMRN vol. 2010, no. 12, 2283–2304.

[Cl1] H. Clemens,Degeneration of Kahler manifolds, Duke Math. J. 44 (1977), 212–290.

[Cl2] ———, The Neron model for families of intermediate Jacobians acquiring“algebraic” singularities, Publ. IHES 58 (1983), 5–18.

[Co] A. Collino,Griffiths’s infinitesimal invariant and higherK-theory on hyperellipticJacobians, J. Alg. Geom. 6 (1997), no.3, 393–415.

[dCM] M. A. de Cataldo and L. Migliorini,On singularities of primitive cohomologyclasses, Proc. Amer. Math. Soc. 137 (2009), 3593–3600.

[DM1] P. del Angel and S. Muller-Stach,Differential equations associated to familiesof algebraic cycles, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2075–2085.

[DM2] ———, Picard–Fuchs equations, integrable systems, and higher algebraicK-theory, in “Calabi–Yau Varieties and Mirror Symmetry” (Yui and Lewis, editors),pp. 43–56, AMS, 2003.

[De1] P. Deligne,Theorie de Hodge II, III, Inst. Hautes Etudes Sci. Publ. Math. 40(1971), 5–57; 44 (1974), 5–77.

[De2] P. Deligne,Hodge cycles on abelian varieties(notes by J. Milne), in “Hodgecycles, motives, and Shimura varieties”, LNM 900, Springer, New York, 1982.

[De3] ———, letter to Cattani and Kaplan (unpublished).

[De4] ———, Local behavior of Hodge structures at infinity, AMS/IP Stud. Adv.Math. 1 (1997), AMS.

[DK] C. Doran and M. Kerr,Algebraic K-theory of toric hypersurfaces, preprint,arXiv:0809.5279.

[EZ] F. El Zein and S. Zucker,Extendability of normal functions associated to alge-braic cycles, in “Topics in transcendental algebraic geometry”, Ann. Math. Stud.106, Princeton Univ. Press, Princeton, NJ, 1984.

[GGM] A. Galligo, M. Granger, P. Maisonobe,D-modules at faisceaux pervers dontle support singulier est un croisement normal, Ann. Inst. Fourier 35 (1985), no. 1,1–48.

[GMV] M. Green, J. Murre, and C. Voisin, “Algebraic cycles and Hodge theory”(Albano and Bardelli, editors), LNM 1594, Springer, 1994.

[Gr1] P. Griffiths,A theorem concerning the differential equations satisfied by normalfunctions attached to algebraic cycles, Amer. J. Math. 101 (1979), no. 1, 94–131.

[Gr2] ———, Poincare and algebraic geometry, Bull. AMS 6 (1982), no. 2, 147–159.

[Gr3] ———, On the periods of certain rational integrals II, Ann. of Math. 90 (1969),no. 2,496–541.


[Gr4] ———, Singularities of admissible normal functions, to appear in “Cycles,motives, and Shimura varieties” (Bombay, 2008).

[GC] P. Griffiths and H. Clemens,The intermediate Jacobian of the cubic threefold,Ann. of Math. (2) 95 (1972), 281–356.

[GG] P. Griffiths and M. Green,Algebraic cycles and singularities of normal func-tions, in “Algebraic cycles and motives”, pp. 206–263, LMS Lect. Not. Ser. 343,Cambridge Univ. Press, Cambridge, 2007.

[GGK] P. Griffiths, M. Green and M. Kerr,Neron models and limits of Abel–Jacobimappings, Compositio Math. 146 (2010), 288–366.

[GH] P. Griffiths and J. Harris, “Principles of algebraic geometry”, Wiley, New York,1978.

[GS] P. Griffiths and W. Schmid,Locally homogeneous complex manifolds, Acta Math.123 (1969), 253–302.

[Ha] F. Hazama,Hodge cycles on certain abelian varieties and powers of specialsurfaces, J. Fac. Sci. Univ. Tokyo, Sect. 1a, 31 (1984), 487–520

[K] M. Kashiwara,A study of variation of mixed Hodge structure, Publ. Res. Inst. Math.Sci. 22 (1986), no. 5, 991–1024.

[KNU1] K. Kato, C. Nakayama and S. Usui, SL.2/-orbit theorem for degeneration ofmixed Hodge structure, J. Algebraic Geom. 17 (2008), 401–479.

[KNU2] ———, Log intermediate Jacobians, Proc. Japan Acad. Ser. A Math. Sci.Volume 86, Number 4 (2010), 73–78.

[KNU3] ———, private communication.

[KU] K. Kato and S. Usui, “Classifying spaces of degenerating polarized Hodgestructures”, Ann. Math. Stud. 169, 2009.

[Ka] N. Katz, “Exponential sums and differential equations”, Ann. of Math. Study 124,Princeton University Press, Princeton, NJ, 1990.

[Ki] J. King, Log complexes of currents and functorial properties of the Abel–Jacobimap, Duke Math. J. 50 (1983), no. 1, 1–53.

[KL] M. Kerr and J. Lewis, The Abel–Jacobi map for higher Chowgroups, II, Invent.Math 170 (2007), no. 2, 355–420.

[KLM] M. Kerr, J. Lewis and S. Muller-Stach,The Abel–Jacobi map for higher Chowgroups, Compositio Math. 142 (2006), no. 2, 374–396.

[KP] M. Kerr and G. Pearlstein, Normal functions and the GHC,preprint, available athttp://www.math.wustl.edu/˜matkerr/KPGHC.pdf.

[KY] B. Klingler and A. Yafaev,The Andre–Oort Conjecture, preprint, available athttp://www.math.jussieu.fr/˜klingler/papiers/KY12.pdf.

[L] S. Lefschetz, “L’Analysis situs et la geometrie algebrique”, Gauthier-Villars, Paris,1924.


[Le1] J. Lewis, “A survey of the Hodge Conjecture” (2nd ed.),CRM Monograph Ser.10, AMS, Providence, RI, 1999.

[Le2] ———, A filtration on the Chow groups of a complex projective variety, Com-positio Math. 128 (2001), no. 3, 299–322.

[Li] D. Lieberman,On the module of intermediate Jacobians, Amer. J. Math. 91 (1969),671–682.

[Ma] Y. I. Manin, Rational points on algebraic curves over function fields, Izv. Akad.Nauk SSSR Ser. Mat. 27 (1963), 1395–1440.

[Mo] D. Morrison, The Clemens–Schmid exact sequence and applications, in “Topicsin transcendental algebraic geometry”, pp. 101–119, Ann. Math. Studies, PrincetonUniv. Press, Princeton, NJ, 1984.

[MW] D. Morrison and J. Walcher,D-branes and normal functions, Adv. Theor. Math.Phys. 13 (2009), no. 2, 553–598.

[Mu] D. Mumford, Families of abelian varieties, in “Algebraic groups and discontinu-ous subgroups” (Boulder, CO, 1965), pp. 347–351, AMS, Providence, RI, 1966.

[Mr] V. K. Murty, Exceptional Hodge classes on certain abelian varieties, Math. Ann.268 (1984), 197–206.

[Ms] G. A. Mustafin,Families of algebraic varieties and invariant cycles, Izv. Akad.Mauk. SSSR Ser. Mat. 49 (1985), no. 5, pp. 948–978.

[Na] I. Nakamura,Relative compactification of the Neron model and its application, in“Complex analysis and algebraic geometry”, pp. 207–225, Iwanami Shoten, Tokyo,1977.

[NS] A. Nemethi and J. Steenbrink,Extending Hodge bundles for abelian variations,Annals of Math. 142 (1996), 1–18.

[Pe1] G. Pearlstein,Variations of mixed Hodge structure, Higgs fields and quantumcohomology,Manuscripta Math. 102 (2000), 269–310.

[Pe2] ———,SL2-orbits and degenerations of mixed Hodge structure, J. DifferentialGeom. 74 (2006), no. 1, 1–67.

[Pe3] ———,Hodge conjecture and the period map at infinity, preprint.

[PS1] C. Peters and J. Steenbrink, “Mixed Hodge structures”, Ergebnisse der Mathe-matik ser. 3 vol. 52, Springer, Berlin, 2008.

[PS2] ———, Monodromy of variations of Hodge structure, Acta Appl. Math. 75(2003), no. 1–3, 183–194.

[P1] H. Poincare, Sur les courbes tracees sur les surfaces algebriques,Ann. Sci. del’ Ecole Norm. Sup. 27 (1910), 55–108.

[P2] ———, Sur les courbes tracees sur une surface algebrique, Sitz. der BerlinerMath. Gesellschaft 10 (1911), 28–55.

[RS] A. Rosenschon and M. Saito,Cycle map for strictly decomposable cycles, Amer.J. Math. 125 (2003), 773–790.


[S1] M. Saito,Admissible normal functions, J. Alg. Geom. 5 (1996), no. 2, 235–276.

[S2] ———, Direct image of logarithmic complexes and infinitesimal invariants ofcycles, in “Algebraic cycles and motives”, vol. 2, pp. 304–318, LMSLect. Not. Ser.344, Cambridge Univ. Press, Cambridge, 2007.

[S3] ———, Cohomology classes of admissible normal functions, arXiv:0904.1593,preprint.

[S4] ———, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333.

[S5] ———, Hausdorff property of the Zucker extension at the monodromyinvariantsubspace, arXiv:0803.2771, preprint.

[S6] ———, Variations of Hodge structures having no cohomologically nontrivialadmissible normal functions, preprint.

[S7] ———,On the theory of mixed Hodge modules, AMS Translations (Series 2) 160(1994), 47–61. Translated from Sugaku.

[S8] ———, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988),no. 6, 849–995.

[SS] M. Saito and C. Schnell,A variant of Neron models over curves, to appear inManuscripta Math.

[Sa] S. Saito,Motives and filtrations on Chow groups, Invent. Math. 125 (1996), 149–196.

[Sc] W. Schmid,Variation of Hodge structure: the singularities of the period mapping,Invent. Math. 22 (1973), 211–319.

[Sl1] C. Schnell,Two observations about normal functions, Clay Math. Proc. 9 (2010),75–79.

[Sl2] ———, Local duality and polarized Hodge modules, arXiv:0904.3480, to appearin Publ. Math. Res. Inst. Sci.

[Sl3] ———, “The boundary behavior of cohomology classes andsingularities ofnormal functions,” Ph.D. Thesis, Ohio State University, 2008. Available atOhioLinkETD.

[Sl4] ———, An alternative construction of the Neron model, preprint, 2009.

[Sn] C. Schoen,Varieties dominated by product varieties, Internat. J. Math. 7 (1996),no. 4, 541–571.

[Sh] T. Shioda,A note on a theorem of Griffiths on the Abel–Jacobi map, Invent. Math.82 (1985), no. 3, 461–465.

[Sp] T. A. Springer, “Linear algebraic groups” (2nd ed.), Birkhauser, Boston, 1998.

[St] J. Steenbrink,Limits of Hodge structures, Inventiones Math. 31 (1976), 229–257.

[SZ] J. Steenbrink and S. Zucker,Variation of mixed Hodge structure, I, Invent. Math.80 (1985), no. 3, 489–542.


[Th] R. Thomas,Nodes and the Hodge conjecture, J. Algebraic Geom. 14 (2005), no.1, 177–185.

[Vo1] C. Voisin, Hodge loci and absolute Hodge classes, Compos. Math. 143 (2007),no. 4, 945–958.

[Vo2] ———, “Hodge theory and complex algebraic geometry” (2vol.), CambridgeStud. Adv. Math. 76, 77, Cambridge Univ. Press, Cambridge, 2007.

[Yo] A. Young, “Complex analytic Neron models for abelian varieties over higherdimensional parameter spaces”, Princeton Univ. Ph.D. thesis.

[Zu1] S. Zucker,The Hodge conjecture for cubic fourfolds, Compositio Math. 34(1977), no. 2, 199–209.

[Zu2] ———, Generalized intermediate Jacobians and the theorem on normal func-tions, Inventiones Math. 33 (1976), no. 2, 185–222.

MATT KERR


WASHINGTON UNIVERSITY IN ST. LOUIS

CUPPLESI HALL

ONE BROOKINGS DRIVE

ST. LOUIS, MO 63130UNITED STATES

[email protected]

GREGORY PEARLSTEIN


M ICHIGAN STATE UNIVERSITY

EAST LANSING, MI 48824UNITED STATES

[email protected]


Motivic characteristic classesSHOJI YOKURA

ABSTRACT. Motivic characteristic classes of possibly singular algebraic vari-eties are homology class versions of motivic characteristics, not classes in theso-called motivic (co)homology. This paper is a survey of them, with emphasison capturing infinitude finitely and on the motivic nature, inother words, thescissor relation or additivity.

1. Introduction

Characteristic classes are usually cohomological objectsdefined on real orcomplex vector bundles. For a smooth manifold, for instance, its characteristicclasses are defined through the tangent bundle. For real vector bundles, Stiefel–Whitney classes and Pontraygin classes are fundamental; for complex vectorbundles, the Chern class is the fundamental one.

When it comes to a non-manifold space, such as a singular realor complex al-gebraic or analytic variety, one cannot talk about its cohomological characteristicclass, unlike the smooth case, because one cannot define its tangent bundle —although one can define some reasonable substitutes, such asthe tangent coneand tangent star cone, which are not vector bundles, but stratified vector bundles.

In the 1960s people started to define characteristic classeson algebraic va-rieties as homological objects — not through vector bundles, but as higher ana-logues of geometrically important invariants such as the Euler–Poincare char-acteristic, the signature, and so on. I suppose that the theory of characteristicclasses of singular spaces starts with Thom’sL-class for oriented PL-manifolds

This is an expanded version of the author’s talk at the workshop “Topology of Stratified Spaces” held atMSRI, Berkeley, from September 8 to 12, 2008.Partially supported by Grant-in-Aid for Scientific Research (No. 21540088), the Ministry of Education,Culture, Sports, Science and Technology (MEXT), and JSPS Core-to-Core Program 18005, Japan.

375

376 SHOJI YOKURA

[Thom], whereas Sullivan’s Stiefel–Whitney classes and the so-called Deligne–Grothendieck conjecture about the existence of Chern homology classes startedthe whole story ofcapturing characteristic classes of singular spaces as natu-ral transformations, more precisely as a natural transformation from a certaincovariant functor to the homology functor.

The Deligne–Grothendieck conjecture seems to be based on Grothendieck’sideas or Deligne’s modification of Grothendieck’s conjecture on aRiemann–Roch type formulaconcerning the constructibleetale sheaves and Chow rings(see [Grot, Part II, note(871), p. 361 ff.]) and was made in its well-known currentform by P. Deligne later. R. MacPherson [M1] gave a positive answer to theDeligne–Grothendieck conjecture and, motivated by this solution, P. Baum, W.Fulton and R. MacPherson [BFM1] further established the singular Riemann–Roch Theorem, which is a singular version of Grothendieck–Riemann–Roch,which is a functorial extension of the celebrated Hirzebruch–Riemann–Roch(abbreviated HRR) [Hi]. HRR is the very origin of the Atiyah–Singer IndexTheorem.

The main results of [BSY1] (announced in [BSY2]) are the following:

� “Motivic” characteristic classes of algebraic varieties, which is a class ver-sion of the motivic characteristic. (Note that this “motivic class” isnota classin the so-called motivic cohomology in algebraic/arithmetic geometry.)

� Motivic characteristic classes in a sense give rise toa unification of threewell-known important characteristic homology classes:

(1) MacPherson’s Chern class transformation [M1] (see also[M2; Schw;BrS]);

(2) Baum, Fulton and MacPherson’s Riemann–Roch transformation [BFM1];

(3) Goresky and MacPherson’sL-homology class (see [GM]), or Cappelland Shaneson’sL-homology class [CS1] (cf. [CS2]).

This unification result can be understood to be good enough toconsider our mo-tivic characteristic classes as a positive solution to the following MacPherson’squestion or comment, written at the end of his survey paper of1973 [M2]:

“It remains to be seen whether there is a unified theory of characteristicclasses of singular varieties like the classical one outlined above.”

The current theory unifies “only three” characteristic classes, but so far itseems to be a reasonble one.

The purpose of this paper is mainly to explain the results from [BSY1] men-tioned above (also see [SY]) with emphasis on the “motivic nature” of motiviccharacteristic classes. In particular, we show that our motivic characteristic classis a very natural class version of the so-called motivic characteristic, just like

MOTIVIC CHARACTERISTIC CLASSES 377

the way A. Grothendieck extended HRR to Grothendieck –Riemann–Roch. Forthat, we go back all the way to the natural numbers, which would be thought ofas the very origin of acharacteristicor characteristic class.

We naıvely start with the counting of finite sets. Then we want to count infi-nite sets as if we are still doing the same way of counting finite sets, and want tounderstand motivic characteristic classes as higher-class versions of this unusual“counting infinite sets”, where infinite sets are complex algebraic varieties. (Theusual counting of infinite sets, forgetting the structure ofa variety at all, lead usinto the mathematics of infinity.) The key is Deligne’s mixedHodge structures[De1; De2], or more generally Saito’s deep theory of mixed Hodge modules[Sa2], etc.

As to mixed Hodge modules (MHM), in [Sch3] Jorg Schurmann gives a verynice introduction and overview about recent developments on the interaction oftheories of characteristic classes and mixed Hodge theory for singular spacesin the complex algebraic context with MHM as a crucial and fundamental key.For example, a study of characteristic classes of the intersection homologicalHodge modules has been done in a series of papers by Sylvain Cappell, AnatolyLibgober, Laurentiu Maxim, Jorg Schurmann and Julius Shaneson [CLMS1;CLMS2; CMS1; CMS2; CMSS; MS1; MS2] (in connection with this last one,see also [Y8]).

The very recent book by C. Peters and J. Steenbrink [PS] seemsto be a mostup-to-date survey on mixed Hodge structures and Saito’s mixed Hodge modules.The Tata Lecture Notes by C. Peters [P] (which is a condensed version of [PS])give a nice introduction to Hodge Theory with more emphasis on the motivicnature.1

2. Preliminaries: from natural numbers to genera

We first consider counting the number of elements of finite sets, i.e., naturalnumbers. LetFSET be the category of finite sets and maps among them. Foran objectX 2 FSET , let

c.X / 2 Z

be the number of the elements ofX , which is usually denoted byjX j (2N) andcalled the cardinal number, or cardinality ofX . It satisfies the following fourproperties on the categoryFSET of finite sets:

(1) X ŠX 0 (bijection or equipotent)÷ c.X /D c.X 0/.(2) c.X /D c.X nY /C c.Y / for Y �X .(3) c.X �Y /D c.X / � c.Y /.

1J. Schurmann informed me of the book [PS] and the lecture [P] at the workshop.

378 SHOJI YOKURA

(4) c.pt/D 1. (Herept denotes one point.)

REMARK 2-1. Clearly these four properties characterize the counting c.X /.Also note that ifc.X / 2Z satisfies (1)–(3) without (4), then we havec.pt/D 0

or c.pt/D1. If c.pt/D0, then it follows from (2) (or (1) and (3)) thatc.X /D0

for any finite setX . If c.pt/ D 1, it follows from (2) thatc.X / is the numberof elements of a finite setX .

REMARK 2-2. When it comes to infinite sets, cardinality still satisfies properties(1)–(4), but the usual rules of computation no longer work. For natural numbers,a2 D a impliesaD 0 or aD 1. But the infinite cardinal@ D c.R/ also has theproperty that@2 D@; in fact, for any natural numbern,

c.Rn/D c.R/ , i.e., @n D @:

This leads into themathematics of infinity.One could still imagine counting on the bigger categorySET of sets, where a

set can be infinite, andc.X / lies in some integral domain. However, one can seethat if for such a counting (1), (2) and (3) are satisfied, it follows automaticallythatc.pt/D 0, contradicting property (4).

In other words: if we consider counting with properties (1)–(3) on the cate-gory SET of all sets, the only possibility is the trivial one:c.X / D 0 for anysetX !

However, if we consider sets having superstructures on the infrastructure(set) and property.1/ is replaced by the invariance of the superstructures, wedo obtain more reasonable countings which are finite numbers; thus we canavoid the mysterious “mathematics of infinity” and extend the usual countingc.X / of finite sets very naturally and naıvely. This is exactly what the Eulercharacteristic, the genus, and many other important and fundamental objects inmodern geometry and topology are all about.

Let us consider the following “topological counting”ctop on the categoryT OP

of topological spaces, which assigns to each topological spaceX a certain inte-ger (or more generally, an element in an integral domain)

ctop.X / 2 Z

such that it satisfies the following four properties, which are exactly the sameas above except for (1):

(1) X ŠX 0 (homeomorphism =T OP- isomorphism)÷ ctop.X /D ctop.X0/,

(2) ctop.X /D ctop.X nY /C ctop.Y / for Y �X (for the moment no condition),(3) ctop.X �Y /D ctop.X / � ctop.Y /,(4) ctop.pt/D 1.


REMARK 2-3. As in Remark 2-1, conditions (1) and (3) imply thatctop.pt/D

0 or 1. If c.pt/ D 0, it follows from (1) and (3) thatctop.X / D 0 for anytopological spaceX . Thus the last condition,c.pt/ D 1, means thatctop isa nontrivial counting. Hence, topological countingctop can be regarded asanontrivial, multiplicative, additive, topological invariant.

PROPOSITION2-4. If such actop exists, then

ctop.R1/D�1; hence ctop.R

n/D .�1/n:

Hence ifX is a finiteC W -complex with�n.X / openn-cells, then

ctop.X /DP

n.�1/n�n.X /D �.X /;

the Euler–Poincare characteristic ofX .

The equalityctop.R1/D�1 can be seen by considering

R1 D .�1; 0/t f0g t .0;1/:

Condition (2) impliesctop.R1/D ctop..�1; 0//C ctop.f0g/C ctop..0;1//, so

�ctop.f0g/D ctop..�1; 0//C ctop..0;1//� ctop.R1/:

SinceR1 Š .�1; 0/Š .0;1/, it follows from (1) and (4) that

ctop.R1/D�ctop.f0g/D�1:

The existence of a countingctop can be shown using ordinary homology/coho-mology theory: symbolically,

topological countingctop W ordinary .co/homology theory.

To be more precise, we use Borel–Moore homology theory [BM],the ho-mology theory with closed supports. For a locally compact Hausdorff spaceX , Borel–Moore homology theoryH BM

� .X IR/ with a ring coefficientR isisomorphic to the relative homology theory of the pair.X c;�/, with X c theone-point compactification ofX and� the one point added toX :

H BM�

.X IR/ŠH�.Xc;�IR/:

Hence, ifX is compact, Borel–Moore homology theory is the usual homologytheory:H BM

�.X IR/DH�.X IR/.

Let K be a field, such asR or C. If the Borel–Moore homologyH BM�

.X IK/

is finite-dimensional — say, ifX is a finite C W -complex — then the Euler–Poincare characteristic�BM using the Borel–Moore homology theory with co-efficient fieldK, namely

�BM .X / WDX

n

.�1/n dimK H BMn .X IK/;

380 SHOJI YOKURA

gives rise to a topological counting�top, because it satisfiesH BMn .Rn;K/D K

andH BMk

.Rn;K/D 0 for k 6D n, and thus

�BM .Rn/D .�1/n:

It turns out that for coefficients in a fieldK, Borel–Moore homology isdual2 as a vector spaceto the cohomology with compact support, namely

H BMp .X IK/D Hom.H p

c .X IK/;K/:

SinceK is a field, we have

H BMp .X IK/ŠH p

c .X IK/

Hence the Euler-Poincare characteristic using Borel–Moore homology�BM .X /

is equal to the Euler-Poincare characteristic using cohomology with compactsupport, usually denoted by�c:

�c.X /DX

i

.�1/i dimK H ic .X IK/:

Since it is quite common to use�c , we have

COROLLARY 2-5. For the category of locally compact Hausdorff spaces,

ctopD �c ;

the Euler–Poincare characteristic using cohomology with compact support.

REMARK 2-6. This story could be retold as follows: There might be many waysof “topologically counting” on the categoryT OP of topological spaces, butthey areall identical to the Euler–Poincare characteristic with compact supportwhen restricted to the subcategory of locally compact Hausdorff spaces withfinite dimensional Borel–Moore homologies. Symbolically speaking,

ctopD �c :

Next consider the following “algebraic counting”calg on the categoryVAR

of complexalgebraic varieties (of finite type overC), which assigns to eachcomplex algebraic varietyX a certain element

calg.X / 2R

in a commutative ringR with unity, such that:

(1) X ŠX 0 (VAR-isomorphism)÷ calg.X /D calg.X0/.

(2) calg.X /D calg.X nY /C calg.Y / for a closed subvarietyY �X .

2For ann-dimensional manifoldM the Poincare duality mapPD W H kc .M / Š Hn�k.M / is an

isomorphism and alsoPD W H k.M /Š H BM

n�k.M / is an isomorphism. Thus they arePoincare dual, but

not dual as vector spaces.


(3) calg.X �Y /D calg.X / � calg.Y /.(4) calg.pt/D 1.

Just likec.X / andctop.X /, the last condition simply means thatcalg is a non-trivial counting.

The real numbersR and in general the Euclidean spaceRn are the most fun-damental objects in the categoryT OP of topological spaces, and the complexnumbersC and in general complex affine spacesCn are the most fundamentalobjects in the categoryVAR of complex algebraic varieties. The decompositionof n-dimensional complex projective space as

Pn D C

0 tC1 t � � � tC

n�1 tCn

implies the following:

PROPOSITION2-7. If calg exists, then

calg.Pn/D 1�yCy2�y3C � � �C .�y/n;

wherey WD �calg.C1/ 2R.

REMARK 2-8. Proposition 2-7 already indicates that there could exist infinitelymany ways — as many as the elementsy — to do algebraic countingcalg on thecategoryVAR of complex algebraic varieties. This is strikingly different fromthe topological countingctop and the original countingc, which are uniquelydetermined. This difference of course lies in the complex structure:

setC topological structureC complex structure.

Here there is no question of consideringR1, so the previous argument show-ing thatctop.R

1/ D �1 does not work. In this sense, we should have used thesymbol calg=C to emphasize the complex structure, instead ofcalg. Since weare dealing with only the category of complex algebraic varieties in this paper,we write justcalg. See Remark 2-11 below for the category of real algebraicvarieties.

The existence of acalg — in fact, of many such ways of algebraically counting —can be shown usingDeligne’s theory of mixed Hodge structures[De1; De2],which comes from the algebraic structure:

setC topological structureC complex structureC algebraic structure:

Then the Hodge–Deligne polynomial

�u;v.X / WDX

i;p;q�0

.�1/i.�1/pCq dimC.GrpF

GrWpCq H ic .X;C//u

pvq

382 SHOJI YOKURA

satisfies the four properties above withRD ZŒu; v� and�y WD calg.C1/D uv,

namely any Hodge–Deligne polynomial�u;v with uv D �y is acalg. Here wepoint out that by Deligne’s work only graded terms withp; q � 0 are nontrivial;otherwise one would have�u;v.X / 2 ZŒu;u�1; v; v�1�.

Similarly one can consider the invariant

calg.X / WD �y;�1 2 ZŒy�;

with calg.C1/D�y.

Here we should note that for.u; v/D .�1;�1/ we have

��1;�1.X /D �c.X /D ctop.X /:

Further, for a smooth compact varietyX , �0;�1.X / is the arithmetic genus,while �1;�1.X / is the signature. These three cases,.u; v/D .�1;�1/, .0;�1/

and.1;�1/, are very important.

algebraic countingcalg: mixed Hodge theory

D ordinary (co)homology theoryCmixed Hodge structures:

REMARK 2-9. (See [DK], for example.) The following description is also fine,but we use the one above in our later discussion on motivic characteristic classes:

calg.Pn/D 1CyCy2Cy3C � � �Cyn;

wherey D calg.C1/ 2 ZŒy�. The Hodge–Deligne polynomial is usually denoted

by E.X Iu; v/ and is defined to be

E.X Iu; v/ WDX

i;p;q�0

.�1/i dimC.GrpF

GrWpCq H ic .X;C//u

pvq:

Thus

�u;v.X /DE.X I �u;�v/:

The reason why we define�u;v.X / to beE.X I �u;�v/ rather thanE.X Iu; v/lies in the definition of Hirzebruch’s generalized Todd class and Hirzebruch’s�y characteristic, which will come below.

The algebraic countingcalg specializes to the topological countingctop. Are thereother algebraic countings that specialize to the Hodge–Deligne polynomial�u;v

(which is sensitive to an algebraic structure)?

CONJECTURE2-10. The answer is negative; in other words, there are no extrastructures other than Deligne’s mixed Hodge structure thatcontribute more tothe algebraic countingcalg of complex algebraic varieties.


REMARK 2-11. In the categoryVAR.R/ of real algebraic varieties, we can ofcourse considercalg=R.R

1/ of the real lineR1; therefore we might be temptedto the hasty conclusion that in the category of real algebraic varieties the topo-logical countingctop, i.e., �c , is sufficient. Unfortunately, the argument forctop.R

1/D�1 does not work in the categoryVAR.R/, becauseR1 and.�1; 0/or .0;1/ are not isomorphic as real algebraic varieties. Even among compactvarieties there do exist real algebraic varieties that are homeomorphic but notisomorphic as real algebraic varieties. For instance (see [MP1, Example 2.7]):

Consider the usualnormal crossingfigure eight curve:

F8D f.x;y/ j y2 D x2�x4g:

The proper transform of F8 under the blowup of the plane at the origin is home-omorphic to a circle, and the preimage of the singular point of F8 is two points.

Next take thetangentialfigure eight curve:

tF8D˚

.x;y/ j�

.xC 1/2Cy2� 1��

.x� 1/2Cy2� 1�

D 0

;

which is the union of two circles tangent at the origin. Here the preimage ofthe singular point is a single point. Therefore, in contrastto the category ofcrude topological spaces, in the category ofreal algebraicvarieties an “algebraiccounting”calg=R.R

1/ is meaningful, i.e., sensitive to the algebraic structure.In-deed, as such a real algebraic countingcalg=R.R

1/ there are

thei-th virtual Betti number ˇi.X / 2 Z

and

the virtual Poincare polynomial ˇt .X /DP

i ˇi.X /ti 2 ZŒt �:

They are both identical to the usual Betti number and Poincare polynomial oncompact nonsingular varieties. For the above two figure eight curves F8 andtF8

we indeed have thatˇt .F8/ 6D ˇt .tF8/:

For more details, see [MP1] and [To3], and see also Remark 4-12.

Finally, in passing, we also mention the following “cobordism” countingccob onthe category of closed oriented differential manifolds or the category of stablyalmost complex manifolds:

(1) X ŠX 0 (cobordant, or bordant)÷ ccob.X /D ccob.X0/.

(2) ccob.X tY /D ccob.X /C ccob.Y /. (Note: in this caseccob.X nY / does notmake sense, becauseX nY has to be a closed oriented manifold.)

(3) ccob.X �Y /D ccob.X / � ccob.Y /.(4) ccob.pt/D 1.

384 SHOJI YOKURA

As in the cases of the previous countings, (1) and (3) implyccob.pt/ D 0 orccob.pt/D 1. It follows from (3) thatccob.pt/D 0 implies thatccob.X /D 0 forany closed oriented differential manifoldsX . Thus the last conditionccob.pt/D

1 means that ourccob is nontrivial. Such a cobordism countingccob is nothingbut a genussuch as the signature, theOA-genus, or the elliptic genus. As inHirzebruch’s book, a genus is usually defined as a nontrivialcounting satisfyingproperties (1), (2) and (3). Thus, it is the same as the one given above.

Here is a very simple problem on genera of closed oriented differentiablemanifolds or stably almost complex manifolds:

PROBLEM 2-12.Determine all genera.

Let Iso.G/n be the set of isomorphism classes of smooth closed (and oriented)pure n-dimensional manifoldsM for G D O (or G D SO), or of puren-dimensional weakly (“D stably”) almost complex manifoldsM for G D U ,i.e.,TM ˚RN

Mis a complex vector bundle (for suitableN , with RM the trivial

real line bundle overM ). Then

Iso.G/ WDM

n

Iso.G/n

becomes a commutative graded semiring with addition and multiplication givenby disjoint union and exterior product, with0 and1 given by the classes of theempty set and one point space.

Let ˝G WD Iso.G/=� be the correspondingcobordism ringof closed (G DO) and oriented (G D SO) or weakly (“D stably”) almost complex manifolds(G D U ) as discussed for example in [Stong]. HereM � 0 for a closed puren-dimensionalG-manifold M if and only if there is a compact pure.nC1/-dimensionalG-manifoldB with boundary@B 'M . This is indeed a ring with�ŒM �D ŒM � for G DO or�ŒM �D Œ�M � for G D SO;U , where�M has theopposite orientation ofM . Moreover, forB as above with@B 'M one has

TBj@B ' TM ˚RM :

This also explains the use of the stable tangent bundle for the definition of astably or weakly almost complex manifold.

The following structure theorems are fundamental (see [Stong, Theorems onp. 177 and p. 110]):

THEOREM 2-13. (1) (Thom)˝SO ˝ Q D Q ŒP2;P4;P6; : : : ;P2n; : : : � is apolynomial algebra in the classes of the complex even dimensional projectivespaces.

(2) (Milnor) ˝U�˝Q D Q ŒP1;P2;P3; : : : ;Pn; : : : � is a polynomial algebra in

the classes of the complex projective spaces.


So, if we consider a commutative ringR without torsion for a genus W˝SO!

R, the genus is completely determined by the value .P2n/ of the cobordismclass of each even-dimensional complex projective spaceP2n. Using this valueone could consider the related generating “function” or formal power seriessuch as

P

n .P2n/xn, or

P

n .P2n/x2n, and etc. In fact, a more interesting

problem is to determine allrigid genera such as the signature� and theA-genus:namely, a genus satisfying the following multiplicativityproperty stronger thanthe product property (3):

(3)rigid : .M / D .F / .B/ for a fiber bundleM ! B with fiber F andcompact connected structure group.

THEOREM2-14.Let log .x/ be the “logarithmic” formal power series inRŒŒx��given by

log .x/ WDX

n

1

2nC 1 .P2n/x2nC1:

The genus is rigid if and only if it is an elliptic genus; i.e., its logarithmlog is an elliptic integral; i.e.,

log .x/DZ x

0

1p

1� 2ıt2C "t4dt

for someı; " 2R.

The “only if” part was proved by S. Ochanine [Oc] and the “if part” was first“physically” proved by E. Witten [Wit] and later “mathematically” proved byC. Taubes [Ta] and also by R. Bott and C. Taubes [BT]. See also B. Totaro’spapers [To2; To4].

cobordism countingccob : Thom’s Theorem

rigid genus = elliptic genus : elliptic integral

The oriented cobordism groupSO above was extended by M. Atiyah [At] toa generalized cohomology theory, i.e., the oriented cobordism theoryMSO�.X /

of a topological spaceX . The theoryMSO�.X / is defined by the so-calledThom spectra: the infinite sequence of Thom complexes given,for a topologicalpair .X;Y / with Y �X , by

MSOk.X;Y / WD limn!1

Œ˙n�k.X=Y /;MSO.n/�:

Here the homotopy groupŒ˙n�k.X=Y /;MSO.n/� is stable.As a covariant or homology-like version ofMSO�.X /, M. Atiyah [At] intro-

duced the bordism theoryMSO�.X / geometrically in quite a simple manner:Let f1 WM1! X , f2 WM2! X be continuous maps from closed orientedn-dimensional manifolds to a topological spaceX . f andg are said to be bordant

386 SHOJI YOKURA

if there exists an oriented manifoldW with boundary and a continuous mapg WW ! X such that

(1) gjM1D f1 andgjM2

D f2, and(2) @W DM1[�M2 , where�M2 is M2 with its reverse orientation.

It turns out thatMSO�.X / is a generalized homology theory and

MSO0.pt/DMSO0.pt/D˝SO :

M. Atiyah [At] also showed Poincare duality for an oriented closed manifoldM of dimensionn:

MSOk.M /ŠMSOn�k.M /:

If we replaceSO.n/ by the other groupsO.n/, U.n/, Spin.n/, we get thecorresponding cobordism and bordism theories.

REMARK 2-15 (ELLIPTIC COHOMOLOGY). Given a ring homomorphism' WMSO�.pt/!R, R is anMSO�.pt/-module and

MSO�.X /˝MSO�.pt/R

becomes “almost” a generalized cohomology theory (one not necessarily satis-fying the Exactness Axiom). P. S. Landweber [L] gave an algebraic criterion(called the Exact Functor Theorem) for it to become a generalized cohomologytheory. Applying this theorem, P. E. Landweber, D. C. Ravenel and R. E. Stong[LRS] showed the following theorem:

THEOREM 2-16. For the elliptic genus WMSO�.pt/DMSO�.pt/D˝!

Z�

12

�

Œı; "�, the following functors are generalized cohomology theories:

MSO�.X /˝MSO�.pt/ Z�

12

�

Œı; "�Œ"�1�;


12

�

Œı; "�Œ.ı2� "/�1�;


12

�

Œı; "�Œ��1�;

where�D ".ı2� "/2.

More generally J. Franke [Fr] showed this:

THEOREM2-17.For the elliptic genus WMSO�.pt/DMSO�.pt/D˝SO!

Z�

12

�

Œı; "�, the functor


12

�

Œı; "�ŒP .ı; "/�1�

is a generalized cohomology theory. HereP .ı; "/ is a homogeneous polynomialof positive degree withdegı D 4, deg"D 8.


The generalized cohomology theory


12

�

Œı; "�ŒP .ı; "/�1�

is calledelliptic cohomology theory. It was recently surveyed by J. Lurie [Lu].It is defined in an algebraic manner, but not in a topologically or geometricallysimpler manner thanK-theory or the bordism theoryMSO�.X /. So, peoplehave been searching for a reasonable geometric or topological construction forelliptic cohomology (cf. [KrSt]).

REMARK 2-18 (MUMBO JUMBO). In the above we see that if you just countpoints of a variety simply as a set, we get infinity unless it isa finite set orthe trivial counting0, but that if we count it “respecting” the topological andalgebraic structures we get a certain reasonable number which is not infinity.Getting carried away, “zeta function-theoretic” formulaesuch as

1C 1C 1C � � �C 1C � � � D �12D �.0/;

1C 2C 3C � � � C nC � � � D� 112D �.�1/;

12C 22C 32C � � �C n2C � � � D 0 D �.�2/;

13C 23C 33C � � �C n3C � � � D 1120D �.�3/

could be considered as based on a counting of infinite sets that respects somekind of “zeta structure” on it, whatever that is. In nature, the equality13C23C

33C� � �Cn3C� � � D 1120

is relevant to theCasimir effect, named after the Dutchphysicist Hendrik B. G. Casimir. (See [Wil, Lecture 7] for the connection.) So,nature perhaps already knows what the zeta structure is. It would be fun, evennonmathematically, to imagine what a zeta structure would be on the naturalnumbersN, or the integersZ or the rational numbersQ, or more generally “zetastructured” spaces or varieties. Note that, like the topological countingctopD�,zeta-theoretical counting (denoted byczeta here) was discovered by Euler!

REMARK 2-19. Regarding “counting”, one is advised to read Baez [Ba1; Ba2],Baez and Dolan [BD], and Leinster [Lein].

3. Motivic characteristic classes

Any algebraic countingcalg gives rise to the following naıve ring homomor-phism to a commutative ringR with unity:

calg W Iso.VAR/!R defined bycalg.ŒX �/ WD calg.X /:

Here Iso.VAR/ is the free abelian group generated by the isomorphism classesŒX � of complex varieties. The additivity relation

calg.ŒX �/D calg.ŒX nY �/C calg.ŒY �/ for any closed subvarietyY �X

388 SHOJI YOKURA

— or, in other words,

calg.ŒX �� ŒY �� ŒX nY �/D 0 for any closed subvarietyY �X ;

induces the following finer ring homomorphism:

calg WK0.VAR/!R defined bycalg.ŒX �/ WD calg.X /:

HereK0.VAR/ is the Grothendieck ring of complex algebraic varieties, whichis Iso.VAR/ modulo the additivity relation

ŒX �D ŒX nY �C ŒY � for any closed subvarietyY �X

(in other words, Iso.VAR/ modded out by the subgroup generated by elementsof the formŒX �� ŒY �� ŒX nY � for any closed subvarietyY �X ).

The equivalence class ofŒX � in K0.VAR/ should be written as,ŒŒX ��, say, butwe just use the symbolŒX � for simplicity.

More generally, lety be an indeterminate and consider the following homo-morphismcalg WD �y WD �y;�1, i.e.,

calg WK0.VAR/! ZŒy� with calg.C1/D�y:

This will be called amotivic characteristic, to emphasize the fact that its domainis the Grothendieck ring of varieties.

REMARK 3-1. In fact, for the categoryVAR.k/ of algebraic varieties over anyfield, the above Grothendieck ringK0.VAR.k// can be defined in the sameway.

What we want to do is an analogue to the way that Grothendieck extendedthe celebrated Hirzebruch–Riemann–Roch Theorem (which was the very begin-ning of the Atiyah–Singer Index Theorem) to the Grothendieck–Riemann–RochTheorem. In other words, we want to solve the following problem:

PROBLEM 3-2. Let R be a commutative ring with unity such thatZ � R, andlet y be an indeterminate. Do there exist some covariant functor} and somenatural transformation(here pushforwards are considered for proper maps)

\ W }. /!H BM�

. /˝RŒy�

satisfying conditions(1)–(3) below?

(1) }.pt/DK0.VAR/.(2) \.pt/D calg, i.e.,

\.pt/D calg W }.pt/DK0.VAR/!RŒy�DH BM�

.pt/˝RŒy�:


(3) For the mapping�X WX!pt to a point, for a certain distinguished element�X 2 }.X / we have

�X �.\.�X //D calg.X / 2RŒy� and �X �.�X /D ŒX � 2K0.VAR/:

}.X /\.X /��! H BM

�.X /˝RŒy�

�X �

?

?

y

?

?

y

�X �

}.pt/DK0.VAR/ ��!\.pt/Dcalg

RŒy�:

(If there exist such} and \, then\.�X / could be called themotivic charac-teristic classcorresponding to the motivic characteristiccalg.X /, just like thePoincare dual of the total Chern cohomology classc.X / of a complex manifoldX corresponds to the Euler–Poincare characteristic:�X �.c.X /\ŒX �/D�.X /.)

A more concrete one for the Hodge–Deligne polynomial (a prototype of thisproblem was considered in [Y5]; cf. [Y6]):

PROBLEM 3-3. LetR be a commutative ring with unity such thatZ�R, and letu; v be two indeterminates. Do there exist a covariant functor} and a naturaltransformation(here pushforwards are considered for proper maps)

\ W }. /!H�BM . /˝RŒu; v�

satisfying conditions(1)–(3) below?

(1) }.pt/DK0.VAR/.(2) \.pt/D �u;v, i.e.,

\.pt/D �u;v W }.pt/DK0.VAR/!RŒu; v�DH BM�

.pt/˝RŒu; v�:

(3) For the mapping�X WX!pt to a point, for a certain distinguished element�X 2 }.X / we have

�X �.].�X //D �u;v.X / 2RŒu; v� and �X �.�X /D ŒX � 2K0.VAR/:

One reasonable candidate for the covariant functor} is the following:

DEFINITION 3-4. (See [Lo2], for example.)The relative Grothendieck groupof X , denoted by

K0.VAR=X /;

is defined to be the free abelian group Iso.VAR=X / generated by isomorphismclassesŒV h

�! X � of morphismsh W V ! X of complex algebraic varieties overX , modulo the additivity relation

ŒVh�!X �D ŒV nZ

hjV nZ

��!X C ŒZhjZ��!X � for any closed subvarietyZ � V I

390 SHOJI YOKURA

in other words, Iso.VAR=X / modulo the subgroup generated by the elementsof the form

ŒVh�! X �� ŒZ

hjZ

��! X �� ŒV nZhjV nZ

��! X �

for any closed subvarietyZ � V .

REMARK 3-5. For the categoryVAR.k/ of algebraic varieties over any field,we can consider the same relative Grothendieck ringK0.VAR.k/=X /.

NOTE 1. K0.VAR=pt/DK0.VAR/.

NOTE 2. K0.VAR=X /3 is a covariant functor with the obvious pushforward:for a morphismf WX ! Y , the pushforward

f� WK0.VAR=X /!K0.VAR=Y /

is defined by

f�.ŒVh�! X �/ WD ŒV

fıh��! Y �:

NOTE 3. Although we do not need the ring structure onK0.VAR=X / in laterdiscussion, the fiber product gives a ring structure on it:

ŒV1

h1

�!X � � ŒV2

h2

�!X � WD ŒV1 �X V2

h1�X h2

��! X �:

NOTE 4. If }.X / D K0.VAR=X /, the distinguished element�X is the iso-morphism class of the identity map:

�X D ŒXidX

��!X �:

If we impose one more requirement in Problems 3-2 and 3-3, we can find theanswer. The newcomer is thenormalization condition(or “smooth condition”)that for nonsingularX we have

\.�X /D c`.TX /\ ŒX �

for a certain normalized multiplicative characteristic classc` of complex vectorbundles. Note thatc` is a polynomial in the Chern classes such that it satisfiesthe normalization condition. Here “normalized” means thatc`.E/D 1 for anytrivial bundle E and “multiplicative” means thatc`.E ˚ F / D c`.E/c`.F /,which is called theWhitney sum formula. In connection with the Whitney sumformula, in the analytic or algebraic context, one asks for this multiplicativityfor a short exact sequence of vector bundles (which splits only in the topologicalcontext):

c`.E/D c`.E0/c`.E00/ for 1!E0!E!E00! 1:

3According to a recent paper by M. Kontsevich (“Notes on motives in finite characteristic”, math.AG/0702206), Vladimir Drinfeld calls an element ofK0.VAR=X / “poor man’s motivic function”.


The normalization condition requirement is natural, in thesense that the otherwell-known/studied characteristic homology classes of possibly singular vari-eties are formulated as natural transformations satisfying such a normalizationcondition, as recalled later. Also, as discussed later (seeConjecture 6-1), thisseemingly strong requirement of the normalization condition could be eventuallydropped.

OBSERVATION 3-6. Let�X W X ! pt be the mapping to a point. It followsfrom the naturality of\ and the normalization condition that

calg.ŒX �/D \�

�X �.ŒXidX

��!X �/�

D�X �

�

\.ŒXidX

��!X �/�

D�X �

�

c`.TX /\ ŒX ��

:

for any nonsingular varietyX . Therefore the normalization condition on non-singular varieties implies that for a nonsingular varietyX the algebraic countingcalg.X / has to be the characteristic number or Chern number [Ful; MiSt]. Thisis another requirement oncalg, but an inevitable one if we want to capture itfunctorially (a la Grothendieck–Riemann–Roch) together with the normalizationcondition above for smooth varieties.

The normalization condition turns out to be essential, and in fact it automaticallydetermines the characteristic classc` as follows, if we consider the bigger ringQ Œy� instead ofZŒy�:

PROPOSITION3-7. If the normalization condition is imposed in Problems3-2and3-3, the multiplicative characteristic classc` with coefficients inQ Œy� hasto be the generalized Todd class, or the Hirzebruch classTy , defined as follows:for a complex vector bundleV ,

Ty.V / WD

rankVY

iD1

�

˛i.1Cy/

1� e�˛i .1Cy/�˛iy

�

where the i are the Chern roots of the vector bundle: c.V /DrankVQ

iD1

.1C˛i/.

PROOF. The multiplicativity ofc` guarantees that ifX andY are smooth com-pact varieties, then

�X �Y �.c`.T .X �Y /\ ŒX �Y �/D �X �.c`.TX /\ ŒX �/ ��Y �.c`.T Y /\ ŒY �/:

In other words, the Chern number is multiplicative, i.e., itis compatible with themultiplicativity of calg. Now Hirzebruch’s theorem [Hi, Theorem 10.3.1] saysthat if the multiplicative Chern number defined by a multiplicative characteristicclassc` with coefficients inQ Œy� satisfies that the corresponding characteristicnumber of the complex projective spacePn is equal to1�yCy2�y3C� � �C

.�y/n, then the multiplicative characteristic classc` has to be the generalizedTodd class, i.e., the Hirzebruch classTy above. ˜

392 SHOJI YOKURA

REMARK 3-8. In other words, in a sensecalg.C1/ uniquely determines the class

version of the motivic characteristiccalg, i.e., the motivic characteristic class.This is very similar to the fact foreseen thatctop.R

1/D�1 uniquely determinesthe “topological counting”ctop.

The Hirzebruch classTy specializes to the following important characteristicclasses:

y D�1 W T�1.V /D c.V /DrankVQ

iD1

.1C˛i/ (total Chern class)

y D 0 W T0.X /D td.V /DrankVQ

iD1

˛i

1� e�˛i

(total Todd class)

y D 1 W T1.X /DL.V /DrankVQ

iD1

˛i

tanh˛i(total Thom–Hirzebruch class)

Now we are ready to state our answer to Problem 3-2, which is one of themain theorems of [BSY1]:

THEOREM 3-9 (MOTIVIC CHARACTERISTIC CLASSES). Let y be an indeter-minate.

(1) There exists a unique natural transformation

Ty�WK0.VAR=X /!H BM

�.X /˝Q Œy�

satisfying the normalization condition that for a nonsingular variety X

Ty�.ŒX

idX

��! X �/D Ty.TX /\ ŒX �:

(2) For X Dpt , the transformationTy�WK0.VAR/!Q Œy� equals the Hodge–

Deligne polynomial

�y;�1 WK0.VAR/! ZŒy��Q Œy�;

namely,

Ty�.ŒX ! pt �/D �y;�1.ŒX �/D

X

i;p�0

.�1/i dimC.GrpF

H ic .X;C//.�y/p:

�y;�1.X / is simply denoted by�y.X /.

PROOF. (1) The main part is of course the existence of such aTy�, the proof of

which is outlined in a later section. Here we point out only the uniqueness ofTy�

, which follows from resolution of singularities. More precisely it followsfrom two results:


(i) Nagata’s compactification theorem, or, if we do not wish to use such a fancyresult, the projective closure of affine subvarieties. We get the surjectivehomomorphism

A W Isoprop.VAR=X /“ K0.VAR=X /;

where Isoprop.VAR=X / is the free abelian group generated by the isomor-phism class ofpropermorphisms toX .

(ii) Hironaka’s resolution of singularities: it implies, by induction on dimensionthat any isomorphism classŒY h

�!X � can be expressed as

X

V

aV ŒVhV

��! X �;

with V nonsingular andhV W V !X proper. We get the surjective maps

Isoprop.SM=X /“ Isoprop.VAR=X /I

therefore

B W Isoprop.SM=X /“ K0.VAR=X /;

where Isoprop.SM=X / is the free abelian group generated by the isomor-phism class ofpropermorphisms fromsmooth varietiesto X .

(iii) The normalization condition (“smooth condition”) ofpage 390.(iv) The naturality ofTy�

.

The two surjective homomorphismsA andB also play key roles in the proof ofthe existence ofTy�

.

(2) As pointed out in (ii),K0.VAR/ is generated by the isomorphism classesof compact smooth varieties. On a nonsingular compact variety X we have

�y;�1.X /DX

p;q�0

.�1/q dimC H q.X I˝pX/yp;

which is denoted by�y.X / and is called the Hirzebruch�y-genus. Next wehave thegeneralized Hirzebruch–Riemann–Roch Theorem(gHRR), which says[Hi] that

�y.X /D

Z

X

Ty.TX /\ ŒX �:

SinceZ

X

Ty.TX /\ ŒX �D �X �.Ty.TX /\ ŒX �/D Ty�.ŒX ! pt �/, we have

Ty�.ŒX ! pt �/D �y;�1.ŒX �/

on generators ofK0.VAR/, and hence on all ofK0.VAR/; thusTy�D �y;�1.

˜

394 SHOJI YOKURA

REMARK 3-10. Problem 3-3 is slightly more general than Problem 3-2 in thesense that it involves two indeterminatesu; v. However, the important keysare the normalization condition for smooth compact varieties and the fact that�u;v.P

1/ D 1C uv C .uv/2 C � � � C .uv/n, which automatically implies thatc`D T�uv, as shown in the proof above. In fact, we can say more aboutu andv: eitheruD�1 or vD�1, as shown below (see also [Jo] — the arXiv version).Hence, we can conclude that for Problem 3-3 there isno such transformation] WK0.VAR=�/!H BM

�.�/˝RŒu; v�with both intermediatesu andv varying.

To show the claim aboutu andv, suppose that forX smooth and for a certainmultiplicative characteristic classc` we have

�u;v.X /D �X �.c`.TX /\ ŒX �/:

In particular, consider a smooth elliptic curveE and anyd-fold covering

� W zE!E

with zE a smooth elliptic curve. Note thatT zE D ��TE and

�u;v.E/D �u;v. QE/D 1CuC vCuv D .1Cu/.1C v/:

Hence we have

.1Cu/.1C v/D �u;v. zE/D � zE�.c`.T zE/\ Œ zE�/D � zE�

.c`.��TE/\ Œ zE�/

D �E��.c`.��TE/\ Œ zE�/D �E�.c`.TE/\��Œ zE�/

D �E�.c`.TE/\ d ŒE�/D d ��E�.c`.TE/\ ŒE�/

D d ��u;v.E/D d.1Cu/.1C v/:

Thus we get.1Cu/.1Cv/D d.1Cu/.1Cv/. Sinced 6D 0, we must have that.1Cu/.1C v/D 0, showing thatuD�1 or v D�1.

REMARK 3-11. Note that�u;v.X / is symmetric inu andv; thus both specialcasesuD�1 andvD�1 give rise to the samec`DTy . It suffices to check thisfor a compact nonsingular varietyX . In fact this follows from the Serre duality.

REMARK 3-12. The heart of the mixed Hodge structure is certainly theexis-tence of the weight filtrationW � and the Hodge–Deligne polynomial, i.e., thealgebraic countingcalg, involves the mixed Hodge structure, i.e., both the weightfiltration W � and the Hodge filtrationF�. However, when one tries to capturecalg functorially, only the Hodge filtrationF� gets involved; the weight filtrationdoes not, as seen in the Hodge genus�y .

DEFINITION 3-13. For a possibly singular varietyX , we call

Ty�.X / WD Ty�

.ŒXidX

��!X �/

theHirzebruch class ofX .


COROLLARY 3-14. The degree of the0-dimensional component of the Hirze-bruch class of a compact complex algebraic varietyX is just the Hodge genus:

�y.X /D

Z

X

Ty�.X /:

This is another singular analogue of the gHRR theorem (�y D Ty), which is ageneralization of the famous Hirzebruch–Riemann–Roch Theorem (which wasfurther generalized to the Grothendieck–Riemann–Roch Theorem):

Hirzebruch–Riemann–Roch:pa.X /D

Z

X

td.TX /\ ŒX �;

with pa.X / the arithmetic genus andtd.V / the original Todd class. Noticingthe above specializations of�y andTy.V /, this gHRR is a unification of thefollowing three well-known theorems:

y D�1 W �.X /D

Z

X

c.X /\ ŒX � (Gauss–Bonnet, or Poincare–Hopf)

y D 0 W pa.X /D

Z

X

td.X /\ ŒX � (Hirzebruch–Riemann–Roch)

y D 1 W �.X /D

Z

X

L.X /\ ŒX � (Hirzebruch’s Signature Theorem)

4. Proofs of the existence of the motivic characteristic classTy�

Our motivic characteristic class transformation


�.X /˝Q Œy�

is obtained as the composite

Ty�D BtdBFM

�.y/ı�mot

y

of the natural transformations

�moty WK0.VAR=X /!G0.X /˝ZŒy�

andBtdBFM

�.y/WG0.X /˝ZŒy�!H BM

� .X /˝Q Œy; .1Cy/�1�:

Here, to describeBtdBFM�.y/

, we need to recall the following Baum–Fulton–MacPherson’s Riemann–Roch or Todd class for singular varieties [BFM1]:

396 SHOJI YOKURA

THEOREM 4-1. There exists a unique natural transformation

tdBFM�

WG0.�/!H BM�

.�/˝Q

such that for a smoothX

tdBFM�

.OX /D td.TX /\ ŒX �:

Here G0.X / is the Grothendieck group of coherent sheaves onX , which is acovariant functor with the pushforwardf� W G0.X / ! G0.Y / for a propermorhphismf WX ! Y defined by

f!.F/DX

j

.�1/j ŒRjf�F �:

Now settdBFM

�.X / WD tdBFM

�.OX /I

this is called the Baum–Fulton–MacPherson Todd class ofX . Then

pa.X /D �.X;OX /D

Z

X

tdBFM�

.X / (HRR-type theorem):

Let

tdBFM�i WG0.X /

tdBFM��!H BM

� .X /˝Qprojection��!H BM

2i .X /˝Q

be thei-th (i.e.,2i-dimensional) component oftdBFM�

. Then the abovetwistedBFM-Todd class transformationor twisted BFM-RR transformation(cf. [Y4])

BtdBFM�.y/

WG0.X /˝ZŒy�!H BM�

.X /˝Q Œy; .1Cy/�1�

is defined byBtdBFM

�.y/WDX

i�0

1

.1Cy/itdBFM

�i :

In this process,�moty WK0.VAR=X /!G0.X /˝ZŒy� is the key. This object

was denoted bymC� in our paper [BSY1] and called themotivic Chern class.In this paper, we use the notation�mot

y to emphasize the following property of it:

THEOREM 4-2 (“MOTIVIC ” �y -CLASS TRANSFORMATION). There exists aunique natural transformation


satisfying the normalization condition that for smoothX

�moty .ŒX

id�!X �/D

dimXX

pD0

Œ˝pX�yp D �y.T

�X /˝ ŒOX �:


Here�y.T�X / D

PdimXpD0 Œ�

p.T �X /�yp and˝ŒOX � W K0.X /Š G0.X / is an

isomorphism for smoothX , i.e., taking the sheaf of local sections.

THEOREM 4-3. The natural transformation

Ty�WD BtdBFM

�.y/ı�mot

y WK0.VAR=X /!H BM� .X /˝Q Œy�

�H�.X /˝Q Œy; .1Cy/�1�

satisfies the normalization condition that for smoothX

Ty�.ŒX

id�!X �/D Ty.TX /\ ŒX �:

Hence such a natural transformation is unique.

REMARK 4-4. Why is the image ofTy�in H BM

�.X /˝Q Œy�? Even though the

target of

BtdBFM�.y/

WG0.X /˝ZŒy�!H�.X /˝Q Œy; .1Cy/�1�

is H BM� .X /˝Q Œy; .1Cy/�1�, the image ofTy�

DBtdBFM�.y/

ı�moty is contained

in H�.X /˝Q Œy�. Indeed, as mentioned, by Hironaka’s resolution of singulari-ties, induction on dimension, the normalization condition, and the naturality ofTy�

, the domainK0.VAR=X / is generated byŒV h�! X � with h proper andV

smooth. Hence

Ty�.ŒV

h�!X �/DTy�

.h�ŒVidV

��!V �/Dh�.Ty�.ŒV

idV

��!V �/2H BM�

.X /˝Q Œy�:

PROOF OFTHEOREM4-3. In [BSY1] we gave a slick way of proving this. Herewe give a nonslick, direct one. LetX be smooth.

BtdBFM�.y/

ı�moty .ŒX

id�! X �/

D BtdBFM�.y/

.�y.˝X //DP

i�0

1

.1Cy/itdBFM

�i .�y.˝X //

DP

i�0

1

.1Cy/i�

tdBFM� .�y.˝X //

�

i

DP

i�0

1

.1Cy/i�

tdBFM� .�y.T

�X /˝ ŒOX �/�

i

DP

i�0

1

.1Cy/i�

ch.�y.T�X //\ tdBFM

�.OX /

�

i

DP

i�0

1

.1Cy/i�

ch.�y.T�X //\ .td.TX /\ ŒX �/

�

i

DP

i�0

1

.1Cy/i

�

dimXQ

jD1

.1Cye� j /dimXQ

jD1

j

1�e� j

�

dimX �i

\ ŒX �:

398 SHOJI YOKURA

Furthermore we have

1

.1Cy/i

�

dimXQ

jD1

.1Cye� j /dimXQ

jD1

j

1�e� j

�

dimX �i

D.1Cy/dimX

.1Cy/i

�

dimXQ

jD1

1Cye� j

1Cy

dimXQ

jD1

j

1�e� j

�

dimX �i

D .1Cy/dimX �i

�

dimXQ

jD1

1Cye� j

1Cy

dimXQ

jD1

j

1�e� j

�

dimX �i

D

�

dimXQ

jD1

1Cye� j

1Cy

dimXQ

jD1

j .1Cy/

1�e� j .1Cy/

�

dimX �i

D

�

dimXQ

jD1

1Cye� j

1Cy� j .1Cy/

1�e� j .1Cy/

�

dimX �i

D

�

dimXQ

jD1

j .1Cy/

1�e� j .1Cy/� j y

�

dimX �i

D�

Ty.TX /�

dimX �i:

ThereforeBtdBFM�.y/

ı�moty .ŒX

id�!X �/D Ty.TX /\ ŒX �. ˜

It remains to show Theorem 4-2. There are at least three proofs, each with itsown advantages.

FIRST PROOF(using Saito’s theory of mixed Hodge modules [Sa1; Sa2; Sa3;Sa4; Sa5; Sa6]).

Even though Saito’s theory is very complicated, this approach turns out tobe useful and for example has been used in recent works of Cappell, Libgober,Maxim, Schurmann and Shaneson [CLMS1; CLMS2; CMS1; CMS2; CMSS;MS1; MS2], related to intersection (co)homology. Here we recall only the in-gredients which we need to define�mot

y :

MHM1 : To X one can associate an abelian category ofmixed Hodge modulesMHM.X /, together with a functorial pullbackf � and pushforwardf! on thelevel of bounded derived categoriesDb.MHM.X // for any (not necessar-ily proper) map. These natural transformations are functors of triangulatedcategories.

MHM2 : Let i W Y ! X be the inclusion of a closed subspace, with opencomplementj W U WD XnY ! X . Then one has forM 2 DbMHM.X / adistinguished triangle

j!j�M !M ! i!i

�MŒ1�! :


MHM3 : For all p 2 Z one has a “filtered De Rham complex” functor of trian-gulated categories

grFp DR WDb.MHM.X //!Dbcoh.X /

commuting with proper pushforward. HereDbcoh.X / is the bounded de-

rived category of sheaves ofOX -modules with coherent cohomology sheaves.Moreover, grFp DR.M /D 0 for almost allp andM 2DbMHM.X / fixed.

MHM4 : There is a distinguished elementQHpt 2MHM.pt/ such that

grF�pDR.QH

X /'˝pXŒ�p� 2Db

coh.X /

for X smooth and pure-dimensional. HereQHXWD ��

XQH

pt for �X WX ! pt

a constant map, withQHpt viewed as a complex concentrated in degree zero.

The transformations above are functors of triangulated categories; thus theyinduce functors even on the level ofGrothendieck groups of triangulated cate-gories, which we denote by the same name. Note that for theseGrothendieckgroupswe have isomorphisms

K0.DbMHM.X //'K0.MHM.X // and K0.D

bcoh.X //'G0.X /

by associating to a complex its alternating sum of cohomology objects.Now we are ready for the transformationsmH and grF��DR. Define

mH WK0.VAR=X /!K0.MHM.X // by mH.ŒVf�!X �/ WD Œf!Q

HV �:

In a senseK0.MHM.X // is like the abelian group of “mixed-Hodge-moduleconstructible functions”, with the class ofQH

Xas a “constant function” onX .

The well-definedness ofmH , i.e., the additivity relation follows from property(MHM2). By (MHM3) we get the following homomorphism commuting withproper pushforward:

grF��

DR WK0.MHM.X //!G0.X /˝ZŒy;y�1�

defined by

grF��

DR.ŒM �/ WDX

p

ŒgrF�pDR.M /� � .�y/p

Then we define our�moty as the composite of these two natural transformations:

�moty WD grF

��DR ımH WK0.VAR=X /

mH��!K0.MHM.X //

grF��DR��! G0.X /˝ZŒy�:

400 SHOJI YOKURA

By (MHM4), for X smooth and pure-dimensional we have

grF��

DR ımH.ŒidX �/D

dimXX

pD0

Œ˝pX� �yp 2G0.X /˝ZŒy� :

Thus we get the unique existence of the “motivic”�y-class transformation�mot

y . ˜

SECOND PROOF(using the filtered Du Bois complexes [DB]). Recall the sur-jective homomorphism

A W Isoprop.VAR=X /“ K0.VAR=X /:

We can describe its kernel as follows:

THEOREM 4-5. K0.VAR=X / is isomorphic to the quotient ofIsopro.VAR=X /

modulo the “acyclicity” relation

Œ?!X �D 0 and Œ zX 0!X �� Œ zZ0!X �D ŒX 0!X �� ŒZ0! X �; (ac)

for any cartesian diagram

zZ0 ��! zX 0

?

?

y

?

?

y

q

Z0i

��! X 0 ��! X ;

with q proper, i a closed embedding, andq W QX 0n QZ0!X 0nZ0 an isomorphism.

For a proper mapX 0! X , consider the filtered Du Bois complex

.˝�

X 0 ;F /;

which has the following properties:

(1) ˝�

X 0 is a resolution of the constant sheafC.(2) grp

F.˝�

X 0/ 2Dbcoh.X

0/.(3) LetDR.OX 0/D˝�

X 0 be the de Rham complex ofX 0 with � being the stupidfiltration. Then there is a filtered morphism

� W .˝�

X 0 ; �/! .˝�

X 0 ;F /:

If X 0 is smooth, this is a filtered quasi-isomorphism.

Note thatG0.X0/ŠK0.D

bcoh.X

0//. Let us define

ŒgrpF.˝�

X 0/� WDX

i

.�1/iH i.grpF.˝�

X 0// 2K0.Dbcoh.X

0//DG0.X0/:


THEOREM 4-6. The transformation


defined by

�moty .ŒX 0

h�!X �/ WD

X

p

h�ŒgrpF.˝�

X 0/�.�y/p

is well-defined and is the unique natural transformation satisying the normal-ization condition that for smoothX

�moty .ŒX

idX

��!X �/D

dimXX

pD0


�X /˝OX :

PROOF. The well-definedness follows from the fact that�moty preserves the

acyclicity relation above [DB]. Then uniqueness follows from resolution of sin-gularities and the normalization condition for smooth varieties. ˜

REMARK 4-7. WhenX is smooth, we have

Œgrp� .˝�

X /�D .�1/p Œ˝pX� !

That is why we need.�y/p, instead ofyp, in the definition of�moty .ŒX 0

h�! X �/.

REMARK 4-8. Wheny D 0, we have the natural transformation

�mot0 WK0.VAR=X /!G0.X / defined by�mot

0 .ŒX 0 h�! X �/D h�Œgr0F .˝

�

X 0/�

satisying the normalization condition that for a smoothX

�mot0 .ŒX

idX

��!X �/D ŒOX �: ˜

THIRD PROOF(using Bittner’s theorem onK0.VAR=X / [Bi]). Recall the sur-jective homomorphism

B W Isoprop.SM=X /“ K0.VAR=X /:

Its kernel is identified by F. Bittner and E. Looijenga as follows [Bi]:

THEOREM 4-9. The groupK0.VAR=X / is isomorphic to the quotient ofIsoprop.SM=X / (the free abelian group generated by the isomorphism classesof proper morphisms from smooth varieties toX ) by the “blow-up” relation

Œ?!X �D 0 and ŒBlY X 0!X �� ŒE!X �D ŒX 0!X �� ŒY !X �; (bl)

402 SHOJI YOKURA

for any cartesian diagram

Ei0

��! BlY X 0

?

?

yq0

?

?

y

q

Yi

��! X 0f

��! X ;

with i a closed embedding of smooth(pure-dimensional) spaces andf WX 0!X

proper. HereBlY X 0!X 0 is the blow-up ofX 0 alongY with exceptional divi-sorE. Note that all these spaces overX are also smooth(and pure-dimensionaland/or quasiprojective, if this is the case forX 0 andY ).

The proof of this theorem requires the Weak Factorization Theorem, due toD. Abramovich, K. Karu, K. Matsuki and J. Włodarczyk [AKMW] (see also[Wlo]). ˜

COROLLARY 4-10. (1) LetB� WVAR=k!AB be a functor from the categoryvar=k of (reduced) separated schemes of finite type overspec.k/ to the cate-gory of abelian groups, which is covariantly functorial for proper morphisms,with B�.?/ WD f0g. Assume we can associate to any(quasiprojective) smoothspaceX 2 ob.VAR=k/ of pure dimension a distinguished element

�X 2 B�.X /

such thath�.�X 0/ D �X for any isomorphismh W X 0 ! X . There exists aunique natural transformation

˚ W Isoprop.SM=�/!B�.�/

satisfying the “normalization” condition that for any smooth X

˚.ŒXidX

��!X �/D �X :

(2) Let B� W VAR=k ! AB and�X be as above and furthermore we assumethat

q�.�BlY X /� i�q0

�.�E/D �X � i�.�Y / 2B�.X /

for any cartesian blow-up diagram as in the above Bittner’s theorem withf D idX . Then there exists a unique natural transformation

˚ WK0.VAR=�/!B�.�/

satisfying the “normalization” condition that for any smooth X

˚.ŒXidX

��!X �/D �X :


We will now use Corollary 4-10(2) to conclude our third proof. Consider thecoherent sheaf p

X2G0.X / of a smoothX as the distinguished element�X of

a smoothX . It follows from M. Gros’s work [Gr] or the recent work of Guillenand Navarro Aznar [GNA] that it satisfies the blow-up relation

q�.˝pBlY X

/� i�q0

�.˝pE/D˝

pX� i�.˝

pY/ 2G0.X /;

which in turn implies a blow-up relation for the�y-class:

q�.�y.˝BlY X //� i�q0

�.�y.˝E//D �y.˝X /� i�.�y.˝Y // 2G0.X /˝ZŒy�:

Therefore Corollary 4-10(2) implies this:

THEOREM 4-11.The transformation


defined by

�moty .ŒX 0

h�!X �/ WD h�

�

X

p�0

Œ˝pX 0 �y

p

�

;

whereX 0 is smooth andh W X 0! X is proper, is well-defined and is a uniquenatural transformation satisying the normalization condition that for smoothX

�moty .ŒX

idX

��!X �/D

dimXX

pD0


�X /˝OX :

REMARK 4-12. The virtual Poincare polynomialˇt (Remark 2-11) for thecategoryVAR.R/ of real algebraic varieties is the unique homomorphism

ˇt WK0.VAR.R//! ZŒt � such that t .R1/D t

andˇt .X / D Pt .X / is the classical or usual topological Poincare polynomialfor compact nonsingular varieties. The proof of the existence ofˇi , thusˇt ,also uses Corollary 4-10(2); see [MP1]. Speaking of the Poincare polynomialPt .X /, we emphasize that this polynoimal cannot be a topological counting atall in the category of topological spaces, simply because the argument in theproof of Proposition 2-4 does not work! The Poincare polynomialPt .X / iscertainly amultiplicativetopological invariant, but not anadditiveone.

REMARK 4-13. The virtual Poincare polynomialˇt WK0.VAR.R//! ZŒt � isthe uniqueextension of the Poincare polynomialPt .X / to arbitrary varieties.Note that if we consider complex algebraic varieties, the virtual Poincare poly-nomial

ˇt WK0.VAR/! ZŒt �

404 SHOJI YOKURA

is equal to the following motivic characteristic, using only the weight filtration:

w�.X /DP

.�1/i dimC

�

GrWq H ic .X;C/

�

tq;

because on any smooth compact complex algebraic varietyX they are all thesame:ˇt .X / D Pt .X / D w�.X /. These last equalities follow from the factthat the Hodge structures onH k.X;Q/ are of pure weightk.

This “weight filtration” motivic characteristicw�.X / is equal to the spe-cialization��t;�t of the Hodge–Deligne polynomial for.u; v/D .�t;�t/. Thisobservation implies that there isno class versionof the complex virtual Poincarepolynomialˇt WK0.VAR/! ZŒt �. In other words, there is no natural transfor-mation

\ WK0.VAR=�/!H BM�

.�/˝ZŒt �

satisfying the conditions that

� if X is smooth and compact, then\.ŒXidX

��! X �/ D c`.TX /\ ŒX � for somemultiplicative characteristic class of complex vector bundles; and

� \.pt/D ˇt WK0.VAR/! ZŒt �.

This is because t .X / D ��t;�t .X / for a smooth compact complex algebraicvarietyX (hence for allX ), and so, as in Remark 3-10, one can conclude that.�t;�t/D .�1;�1/. Thust has to be equal to1 and cannot be allowed to vary.In other words, the only chance for such a class version is when t D 1, whichgives the Euler–Poincare characteristic� WK0.VAR/! Z. In that case, we dohave the Chern class transformation

c� WK0.VAR=�/!H BM�

.�IZ/:

This follows again from Corollary 4-10(2) and the blow-up formula of Chernclass [Ful].

REMARK 4-14. The same discussion as in Remark 4-13 can be applied to thecontext of real algebraic varieties, i.e., the same examplefor real elliptic curvesleads us to the conclusion thatt D 1 for ˇt satisfying the corresponding nor-malization condition for a normalized multiplicative characteristic class. Thisclass has to be a polynomial in the Stiefel–Whitney classes,and we end up withthe Stiefel–Whitney homology classw�, which also satisfies the correspondingblow-up formula.

REMARK 4-15 (POOR MAN’ S MOTIVIC CHARACTERISTIC CLASS). If we usethe much simpler covariant functor Isoprop.SM=X / above (the abelian group of“poor man’s motivic functions”), we can get the following “poor man’s motiviccharacteristic class” for any characteristic classc` of vector bundles: Letc` be


anycharacteristic class of vector bundles with coefficient ring K. There existsa unique natural transformation

c`� W Isoprop.SM=�/!H BM� .�/˝K

satisfying the normalization condition that for any smoothvarietyX ,

c`�.ŒXidX

��! X �/D c`.TX /\ ŒX �:

There is a bivariant theoretical version of Isoprop.SM=X / (see [Y7]); a goodreference for it is Fulton and MacPherson’s AMS memoir [FM].

5. Chern class, Todd class and L-class of singular varieties:towards a unification

Our next task is to describe another main theorem of [BSY1], to the effect thatour motivic characteristic classTy�

is, in a sense, a unification of MacPherson’sChern class, the Todd class of Baum, Fulton, and MacPherson (discussed in theprevious section), and the L-class of singular varieties ofCappell and Shaneson.Let’s briefly review these classes:

MacPherson’s Chern class[M1]


cMac�W F.�/!H BM

�.�/

such that, for smoothX ,

cMac� .1X /D c.TX /\ ŒX �:

HereF.X / is the abelian group of constructible functions, which is a covariantfunctor with the pushforwardf� W F.X /! F.Y / for a proper morphismf WX ! Y defined by

f�.1W /.y/D �c.f�1.y/\W /:

We callcMac� .X / WDcMac

� .1X / the MacPherson’s Chern class ofX , or the Chern–Schwartz–MacPherson class. We have

�.X /D

Z

X

cMac�.X /:

406 SHOJI YOKURA

The Todd class of Baum, Fulton, and MacPherson[BFM1]


tdBFM�

WG0.�/!H BM�

.�/˝Q


tdBFM�

.OX /D td.TX /\ ŒX �:

Here G0.X / is the Grothendieck group of coherent sheaves onX , which is acovariant functor with the pushforwardf� W G0.X / ! G0.Y / for a propermorphismf WX ! Y defined by

f!.F/DX

j

.�1/j ŒRjf�F �:

We calltdBFM�

.X / WD tdBFM�

.OX / the Baum–Fulton–MacPherson Todd classof X , and we have

pa.X /D �.X;OX /D

Z

X

tdBFM�

.X /:

TheL-class of Cappell and Shaneson[CS1; Sh] (cf. [Y4])


LCS� W˝.�/!H BM

� .�/˝Q


LCS�.ICX /DL.TX /\ ŒX �:

Here˝.X / is the abelian group of Youssin’s cobordism classes of self-dualconstructible complexes of sheaves onX .

We callLGM�

.X / WD LCS�.ICX / the Goresky–MacPherson homologyL-class

of X . The Goresky–MacPherson theorem [GM] says that

�GM .X /D

Z

X

LGM�

.X /:

We now explain in what sense our motivic characteristic class transformation


�.X /˝Q Œy�

unifies these three characteristic classes of singular varieties, providing a kind ofpartial positive answer to MacPherson’s question4 of whether there is a unifiedtheory of characteristic classes of singular varieties.

4Posed in his survey talk [M2] at the Ninth Brazilian Mathematics Colloquium in 1973.


THEOREM 5-4 (UNIFIED FRAMEWORK FORCHERN, TODD AND HOMOLOGY

L-CLASSES OF SINGULAR VARIETIES).

yD�1: There exists a unique natural transformation" WK0.VAR=�/!F.�/

such that, for X nonsingular, ".Œid W X ! X �/ D 1X , and the followingdiagram commutes:

K0.VAR=X /

T�1� ((Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

"// F.X /

cMac� ˝Qxxpp

p

p

p

p

p

p

p

p

p

H BM� .X /˝Q

yD 0: There exists a unique natural transformation WK0.VAR=�/!G0.�/

such that, for X nonsingular, .Œid W X ! X �/ D ŒOX �, and the followingdiagram commutes:

K0.VAR=X /

T0� ((Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

// G0.X /

tdBFM�wwpp

p

p

p

p

p

p

p

p

p

H BM�

.X /˝Q

yD1: There exists a unique natural transformationsd WK0.VAR=�/!˝.�/

such that, for X nonsingular, sd.Œid W X ! X �/ D ŒQX Œ2 dimX ��, and thefollowing diagram commutes:

K0.VAR=X /

T1� ((Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

sd// ˝.X /

LCS�wwpp

p

p

p

p

p

p

p

p

p

H BM�

.X /˝Q :

The first two claims are straightforward; the third, the casey D 1, is anythingbut. In particular, the existence ofsd W K0.VAR=�/! ˝.�/ is not obviousat all. The only way we know to prove it is by going through somedetailsof Youssin’s work [You] and using Corollary 4-10(2) again. This is done in[BSY1]; see also [BSY2; SY].

REMARK 5-5. y D�1: T�1�.X /D cMac�.X /˝Q.

y D 0: In general, for a singular varietyX we have

�mot0 .ŒX

idX

��!X �/ 6D ŒOX �:

Therefore, in general,T0�.X / 6D tdBFM�

.X /. So, ourT0�.X / shall be calledthe Hodge–Todd class and denoted bytdH

� .X /. However, ifX is a Du Bois

408 SHOJI YOKURA

variety, i.e., every point ofX is a Du Bois singularity (note a nonsingularpoint is also a Du Bois singularity), we DO have

�mot0 .ŒX

idX

��!X �/D ŒOX �:

This is because of the definition of Du Bois variety:X is called a Du Boisvariety if we have

OX D gr0� .DR.OX //Š gr0F .˝�

X /:

Hence, for a Du Bois varietyX we haveT0�.X /D tdBFM�

.X /. For example,S. Kovacs [Kov] proved Steenbrink’s conjecture that rational singularities areDu Bois, thus for the quotientX of any smooth variety acted on by a finitegroup we have thatT0�.X /D tdBFM

�.X /.

yD1: In general,sd.ŒXidX

��!X �/ is distinct fromICX , soT1�.X / 6DLGM�

.X /.We therefore callT1�.X / theHodgeL-classand denote it, alternatively, byLH

�.X /. It is conjectured thatT1�.X / 6D LGM

�.X / for a rational homology

manifoldX .

6. A few more conjectures

CONJECTURE6-1. Any natural transformation

T WK0.VAR=X /!H BM�

.X /˝Q Œy�

without the normalization condition is a linear combination of components ofthe formtdy�i

WK0.VAR=X /!H BM2i

.X /˝Q Œy�:

T DX

i�0

ri.y/ tdy�i.ri.y/ 2Q Œy�/:

This conjecture means that the normalization condition forsmooth varieties im-posed to get our motivic characteristic class can be basically dropped. Thisconjecture is motivated by the following theorems:

THEOREM 6-2 [Y1]. Any natural transformation

T WG0.�/!H BM� .�/˝Q

without the normalization condition is a linear combination of components

tdBFM� i WG0.�/!H BM

2i .�/˝Q;

that is,

T DX

i�0

ri tdBFM� i .ri 2Q/:


THEOREM 6-3 [KMY]. Any natural transformation

T W F.�/!H BM�

.�/˝Q

without the normalization condition is a linear combination of components

cMac� i ˝Q WG0.�/!H BM

2i .�/˝Q

of therationalizedMacPherson’s Chern classcMac� ˝Q (i.e., a linear combina-

tion of cMac� i mod torsion):

T DX

i�0

ri cMac� i ˝Q .ri 2Q/:

REMARK 6-4. This theorem certainly implies the uniqueness of such atransfor-mationcMac

�˝Q satisfying the normalization. The proof of Theorem 6-3does

not appeal to the resolution of singularities at all, thereforemodulo torsion theuniqueness of the MacPherson’s Chern class transformationcMac

�is proved with-

out using resolution of singularities. However, in the caseof integer coefficients,as shown in [M1], the uniqueness ofcMac

�uses the resolution of singualrities and

as far as the author knows, there is no proof available without using this result.Does there exist any mysterious connection between resolution of singularitiesand finite torsion? (In this connection we quote a comment by J. Schurmann:

There is indeed a relation between resolution of singularities and torsioninformation: in [To1] B. Totaro shows by resolution of singularities thatthe fundamental classŒX � of a complex algebraic varietyX lies in the im-age from the complex cobordismU .X /!H�.X;Z/. And this impliessome nontrivial topological restrictions: for example, all odd-dimensionalelements of the Steenrod algebra vanish onŒX � viewed inH�.X;Zp/.)

Furthermore, hinted by these two theorems, it would be natural to speculate thefollowing “linearity” on the Cappell–ShanesonL-class also:

CONJECTURE6-5. Any natural transformation without the normalization con-dition

T W˝.�/!H BM� .�/˝Q

is a linear combination of componentsLCS� i W˝.�/!H BM

2i.�/˝Q:

T DX

i�0

ri LCS� i .ri 2Q/:

410 SHOJI YOKURA

7. Some more remarks

For complex algebraic varieties there is another importanthomology the-ory. That is Goresky–MacPherson’sintersection homology theoryIH , intro-duced in [GM] (see also [KW]). It satisfies all the propertieswhich the ordinary(co)homology theory for nonsingular varieties have, in particular the Poincareduality holds, in contrast to the fact that in general it fails for the ordinary(co)homology theory of singular varieties. In order that the Poincare dualitytheorem holds, one needs to control cycles according toperversity, which issensitive to, or “control”, complexity of singularities. M. Saito showed thatIH

satisfies pure Hodge structure just like the cohomology satisfies the pure Hodgestructure for compact smooth manifolds (see also [CaMi1; CaMi2]). In thissense,IH is a convenient gadget for possibly singular varieties, andusing theIH , we can also get various invariants which are sensitive to the structure ofgiven possibly singular varieties. For the history ofIH , see Kleiman’s surveyarticle [Kl], and forL2-cohomology— very closely related to the intersectionhomology — see [CGM; Go; Lo1; SS; SZ], for example. Thus for the categoryof compact complex algebraic varieties two competing machines are available:

ordinary (co)homologyC mixed Hodge structures

intersection homologyC pure Hodge structures

Of course, they are the same for the subcategory of compact smooth varieties.So, for singular varieties one can introduce the similar invariants usingIH ;

in other words, one can naturally think of theIH -version of the Hirzebruch�y genus, because of the pure Hodge structure, denote by�IH

y : Thus we haveinvariants�y-genus and�IH

y -genus. As to the class version of these, one shouldgo through the derived category of mixed Hodge modules, because the intersec-tion homology sheaf lives in it. Then obviously the difference between these twogenera or between the class versions of these two genera should come from thesingularities of the given variety. For this line of investigation, see the articles byCappell, Libgober, Maxim, and Shaneson [CMS1; CMS2; CLMS1;CLMS2].

The most important result is theDecomposition Theoremof Beilinson, Bern-stein, Deligne, and Gabber [BBD], which was conjectured by I. M. Gelfand andR. MacPherson. A more geometric proof of this is given in the above mentionedpaper [CaMi1] of M. de Cataldo and L. Migliorini.

Speaking of the intersection homology, the general category for IH is thecategory of pseudomanifolds and the canonical and well-studied invariant forpseudomanifolds is the signature, because of the Poincare duality ofIH . Ba-nagl’s monograph [Ba1] is recommended on this topic; see also [Ba2; Ba3;Ba4; BCS; CSW; CW; Wei]. Very roughly,Ty�

is a kind of deformation or


perturbation of Baum–Fulton–MacPherson’s Riemann–Roch.It would be inter-esting to consider a similar kind of deformation ofL-class theory defined onthe (co)bordism theory of pseudomanifolds. Again we quote J. Schurmann:

A deformation of theL-class theory seems not reasonable. Only the signa-ture =�1-genus factorizes over the oriented cobordism ring˝SO , so thatthis invariant is of more topological nature related to stratified spaces. Forthe other desired (“deformation”) invariants one needs a complex algebraicor analytic structure. So what is missing up to now is a suitable theory ofalmost complex stratified spaces.

Finally, since we started the present paper with counting, we end with posingthe following question: how about counting pseudomanifolds respecting thestructure of pseudomanifolds:

Does “stratified counting”cstra make sense?

For complex algebraic varieties, which are pseudomanifolds, algebraic count-ing calg (using mixed Hodge theory = ordinary (co)homology theory + mixedHodge structure) in fact ignores the stratification. So, in this possible problem,one should consider intersection homology + pure Hodge structure, althoughintersection homologyis a topological invariant, and hence independent of thestratification.

J. Schurmann provides one possible answer to the highlighted question above:

One possible answer would be to work in the complex algebraiccontextwith a fixed (Whitney) stratificationX�, so that the closure of a stratumS is a union of strata again. Then one can work with the GrothendieckgroupK0.X�/ of X�-constructible sets, i.e., those which are a union ofsuch strata. The topological additive counting would be related again tothe Euler characteristic and the groupF.X�/ of X�-constructible functions.A more sophisticated version is the Grothendieck groupK0.X�/ of X�-constructible sheaves (or sheaf complexes). These are generated by classesj!LS for j W S !X , the inclusion of a stratumS , andLS a local systemon S , and also by the intermediate extensionsj!�LS , which are perversesheaves. In relation to signature and duality, one can work with the cor-responding cobordism group.X�/ of Verdier self-dualX�-constructiblesheaf complexes. These are generated byj!�LS , with LS a self-dual lo-cal system onS . Finally one can also work with the Grothendieck groupK0.MHM.X�// of mixed Hodge modules, whose underlying rational com-plex is X�-constructible. This last group is of course not a topologicalinvariant.

We hope to come back to the problem of a possible “stratified counting” cstra.

412 SHOJI YOKURA

Acknowledgements

This paper is based on my talk at the workshop “Topology of StratifiedSpaces” held at MSRI, Berkeley, from September 8 to 12, 2008.I thank theorganizers (Greg Friedman, Eugenie Hunsicker, Anatoly Libgober, and Lauren-tiu Maxim) for their invitation to the workshop. I also thankthe referee andJorg Schurmann for their careful reading of the paper and valuable commentsand suggestions, and G. Friedmann and L. Maxim for their much-appreciatedfeedback on an earlier version of the paper. Finally I thank also Silvio Levy,the editor of MSRI Book Series, for his careful reading and editing of the finalversion.

References

[AKMW] D. Abramovich, K. Karu, K. Matsuki and J. Włodarczyk,Torification andfactorization of birational maps, J. Amer. Math. Soc. 15:3 (2002), 531–572.

[At] M. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200–208.

[Ba1] J. Baez,Euler Characteristic versus homotopy cardinality, lecture at the FieldsInstitute Program on Homotopy Theory and its Applications (London, ON, 2003),http://math.ucr.edu/home/baez/cardinality/.

[Ba2] J. Baez,The mysteries of counting: Euler characteristic versus homotopy car-dinality, public lecture at conference “Categories in Algebra, Geometry and Mathe-matical Physics”, (Sydney, 2005), http://math.ucr.edu/home/baez/counting/.

[BD] J. Baez and J. Dolan,From finite sets to Feynman diagrams, in “Mathematicsunlimited: 2001 and beyond”, vol. 1, edited by B. Engquist and W. Schmid, Springer,Berlin, 2001, 29–50.

[Ba1] P. Balmer,Derived Witt groups of a scheme, J. Pure Appl. Algebra 141 (1999),101–129.

[Ba1] M. Banagl,Topological invariants of stratified spaces, Springer, Berlin, 2007.

[Ba2] M. Banagl,Extending intersection homology type invariants to non-Witt spaces,Memoirs Amer. Math. Soc. no. 760 (2002), 1–83.

[Ba3] M. Banagl,Computing twisted signatures and L-classes of non-Witt spaces, Proc.London Math. Soc. (3) 92 (2006), 428–470.

[Ba4] M. Banagl,The L-class of non-Witt spaces, Ann. Math. 163:3 (2006), 743–766.

[BCS] M. Banagl, S. E. Cappell and J. L. Shaneson,Computing twisted signatures andL-classes of stratified spaces, Math. Ann. 326 (2003), 589–623.

[BFM1] P. Baum, W. Fulton and R. MacPherson,Riemann–Roch for singular varieties,Publ. Math. IHES 45 (1975), 101–145.

[BBD] A. A. Beilinson, J. N. Bernstein and P. Deligne,Faisceaux pervers, Asterisque100 (1982),


[Bi] F. Bittner, The universal Euler characteristic for varieties of characteristic zero,Comp. Math. 140 (2004), 1011–1032.

[BM] A. Borel and J. C. Moore, Homology theory for locally compact spaces, Michi-gan Math. J. 7 (1960), 137–159.

[BL1] L. Borisov and A. Libgober,Elliptic genera for singular varieties, Duke Math.J. 116 (2003), 319–351.

[BL2] L. Borisov and A. Libgober,McKay correspondence for elliptic genera, Ann.Math. 161 (2005), 1521–1569,

[BL3] L. Borisov and A. Libgober,Higher elliptic genera, Math. Res. Lett. 15 (2008),511–520.

[BT] R. Bott and C. Taubes,On the rigidity theorems of Witten, J. Amer. Math. Soc. 2(1989), 137–186.

[BrS] J.-P. Brasselet and M.-H. Schwartz,Sur les classes de Chern d’une ensembleanalytique complexe, pp. 93–148 in Caracteristique d’Euler–Poincare, Seminaire E.N. S. 1978–1979, Asterisque 82–83 (1981).

[BSY1] J.-P. Brasselet, J. Schurmann and S. Yokura,Hirzebruch classes and motivicChern classes for singular spaces, J. Topol. Anal. 2:1 (2010), 1–55.

[BSY2] J.-P. Brasselet, J. Schurmann and S. Yokura,Classes de Hirzebruch et classesde Chern motiviques, C. R. Acad. Sci. Paris Ser. I 342 (2006), 325–328.

[CLMS1] S. E. Cappell, A. Libgober, L. G. Maxim and J. L. Shaneson,Hodge generaof algebraic varieties, II, Math. Ann. 345:4 (2009), 925–972.

[CLMS2] S. E. Cappell, A. Libgober, L. G. Maxim and J. L. Shaneson,Hodge generaand characteristic classes of complex algebraic varieties, Electron. Res. Announc.Math. Sci. 15 (2008), 1–7.

[CMSS] S. E. Cappell, L. G. Maxim, J. Schurmann and J. L. Shaneson,Characteristicclasses of complex hypersurfaces, Adv. Math. 225:5 (2010), 2616–2647.

[CMS1] S. E. Cappell, L. G. Maxim and J. L. Shaneson,Euler characteristics ofalgebraic varieties, Comm. Pure Appl. Math. 61:3, (2008), 409–421.

[CMS2] S. E. Cappell, L. G. Maxim and J. L. Shaneson,Hodge genera of algebraicvarieties, I, Comm. Pure Appl. Math. 61:3, (2008), 422–449.

[CS1] S. E. Cappell and J. L. Shaneson,Stratifiable maps and topological invariants,J. Amer. Math. Soc. 4 (1991), 521–551.

[CS2] S. E. Cappell and J. L. Shaneson,Genera of algebraic varieties and countinglattice points, Bull. Amer. Math. Soc. 30 (1994), 62–69.

[CSW] S. E. Cappell, J. L. Shaneson and S. Weinberger,Classes topologiques car-acteristiques pour les actions de groupes sur les espaces singuliers, C. R. Acad. Sci.Paris Ser. I. Math. 313 (1991), 293–295.

[CW] S. E. Cappell and S. Weinberger,Classification de certains espaces stratifies, C.R. Acad. Sci. Paris Ser. I. Math. 313 (1991), 399–401.

414 SHOJI YOKURA

[CGM] J. Cheeger, M. Goresky and R. MacPherson,L2 cohomology and intersectionhomology for singular algebraic varieties, in Seminar on differential geometry,Annals Math. Studies 102 (1982), 303–340

[DK] V. I. Danilov and A. G. Khovanskii,Newton polyhedra and an algorithm forcomputing Hodge–Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50:5 (1986),925–945; In Russian; translated in Bull. Amer. Math. Soc. 30(1994), 62–69.

[CaMi1] M. de Cataldo and L. Migliorini,The Hodge theory of algebraic maps, Ann.Scient.Ec. Norm. Sup. (4) 38 (2005), 693–750.

[CaMi2] M. de Cataldo and L. Migliorini,The Decomposition Theorem and the topol-ogy of algebraic maps, to appear in Bull. Amer. Math. Soc.

[De1] P. Deligne,Theorie de Hodge II, Publ. Math. IHES 40 (1971), 5–58.

[De2] P. Deligne,Theorie de Hodge III, Publ. Math. IHES 44 (1974), 5–78.

[DB] Ph. Du Bois,Complexe de de Rham filtre d’une variete singuliere, Bull. Soc.Math. France 109 (1981), 41–81.

[Fr] J. Franke,On the construction of elliptic cohomology, Math. Nachr. 158 (1992),43–65.

[Fri] G. Friedman,Intersection homology of stratified fibrations and neighborhoods,Adv. in Math. 215 (2007), 24–65.

[Ful] W. Fulton,Intersection theory, Springer (1984).

[FL] W. Fulton and S. Lang,Riemann–Roch algebra, Springer (1985).

[FM] W. Fulton and R. MacPherson,Categorical framework for the study of singularspaces, Mem. Amer. Math. Soc. 243 (1981).

[FuM] J. Fu and C. McCrory,Stiefel–Whitney classes and the conormal cycle of asingular variety, Trans. Amer. Math. Soc. 349 (1997), 809–835.

[Go] M. Goresky,L2-cohomology is intersection cohomology, preprint, available athttp://www.math.ias.edu/�goresky/pdf/zucker.pdf.

[GM] M. Goresky and R. MacPherson,Intersection homology theory, Topology 149(1980), 155–162.

[Gr] M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge–Wittlogarithmique, Bull. Soc. Math. France Mem. 21 (1985).

[Grot] A. Grothendieck,Recoltes et semailles: reflexions et temoignages sur un passede mathematicien, preprint, 1985.

[GNA] F. Guill en and V. Navarro Aznar,Un critere d’extension des foncteurs definissur les schemas lisses, Publ. Math. IHES 95 (2002), 1–91.

[Hi] F. Hirzebruch,Topological methods in algebraic geometry, Springer, New York(1966).

[HBJ] F. Hirzebruch, T. Berger and R. Jung,Manifolds and modular forms, AspectsMath. E.20, Vieweg (1992).


[Jo] D. Joyce,Constructible functions on Artin stacks, math. AG/0403305v4. A revisedversion appeared in J. London Math. Soc. 74 (2006), 583–606.

[KS] M. Kashiwara and P. Schapira,Sheaves on manifolds, Springer, Berlin (1990).

[Ken] G. Kennedy,MacPherson’s Chern classes of singular varieties, Com. Algebra.9 (1990), 2821–2839.

[KMY] G. Kennedy, C. McCrory and S. Yokura,Natural transformations from con-structible functions to homology, C. R. Acad. Sci. Paris I 319 (1994), 969–973,

[KW] F. Kirwan and J. Woolf,An introduction to intersection homology theory(2nded.), Chapman Hall/CRC, 2006.

[Kl] S. Kleiman, The development of intersection homology theory, in “A century ofmathematics in America”, Part II, Hist. Math. 2, Amer. Math.Soc., (1989), 543–585.

[Kov] S. J. Kovacs,Rational, log canonical, Du Bois singularities: on the conjecturesof Kollar and Steenbrink, Comp. Math. 118 (1999), 123–133.

[KrSt] M. Kreck and S. Stolz,HP2-bundles and elliptic homology, Acta Math., 171(1993), 231–261.

[L] P. S. Landweber,Homological properties of comodules overMU�MU andBP�BP,Amer. J. Math., 98 (1976), 591–610.

[LRS] P. S. Landweber, D. C. Ravenel and R. E. Stong,Periodic cohomology theoriesdefined by elliptic curves, Contemp. Math., 181 (1995), 317–337.

[Lein] T. Leinster,The Euler characteristic of a category, Documenta Mathematica 13(2008), 21–49.

[LM] M. Levine and F. Morel,Algebraic cobordism, Springer (2006).

[Lo1] E. Looijenga,L2-cohomology of locally symmetric varieties, Compositio Math.67:1 (1988), 3–20.

[Lo2] E. Looijenga,Motivic measures, Seminaire Bourbaki 874, Asterisque 276 (2002),267–297.

[Lu] J. Lurie, A survey of elliptic cohomology, available at http://www-math.mit.edu/~lurie/papers/survey.pdf, 2007.

[M1] R. MacPherson,Chern classes for singular algebraic varieties, Ann. Math. 100(1974), 423–432.

[M2] R. MacPherson,Characteristic classes for singular varieties, Proceedings of the9th Brazilian Mathematical Colloquium (Pocos de Caldas, 1973) Vol. II, Instituto deMatematica Pura e Aplicada, Rio de Janeiro, (1977), 321–327.

[MS1] L. Maxim and J. Schurmann,Hodge-theoretic Atiyah–Meyer formulae andthe stratified multiplicative property, in “Singularities I: Algebraic and AnalyticAspects”, Contemporary Math. 474 (2008), 145–167.

[MS2] L. Maxim and J. Schurmann,Twisted genera of symmetric products, preprint,arXiv:0906.1264v1.

416 SHOJI YOKURA

[MP1] C. McCrory and A. Parusinski,Virtual Betti numbers of real algebraic varieties,C. R. Acad. Sci. Paris, Ser. I 336 (2003), 763–768.

[MP2] C. McCrory and A. Parusinski,The weight filtration for real algebraic varieties,in this volume.

[Mi-Ra] Elliptic cohomology: geometry, applications, and higher chromatic analogues,edited by H. R. Miller and D. Ravenel, London Math. Soc. Lecture Note Series 342(2007).

[MiSt] J. W. Milnor and J. D. Stasheff,Characteristic classes, Ann. Math. Studies 76,Princeton Univ. Press (1974).

[Oc] S. Ochanine,Sur les genres multiplicatifs definis par des integrales elliptiques,Topology 26 (1987), 143–151.

[P] C. A. M. Peters,Tata Lectures on motivic aspects of Hodge theory, Lecture Notes,Tata Institute, Mumbai, December 2007.

[PS] C. A. M. Peters and J. H. M. Steenbrink,Mixed Hodge structures, Springer, Berlin,2008.

[Sa1] M. Saito,Modules de Hodge polarisables, Publ. RIMS 24 (1988), 849–995.

[Sa2] M. Saito,Mixed Hodge Modules, Publ. RIMS 26 (1990), 221–333.

[Sa3] M. Saito,Induced D-modules and differential complexes, Bull. Soc. Math. France117 (1989), 361–387.

[Sa4] M. Saito,D-modules on analytic spaces, Publ. RIMS 27 (1991), 291–332.

[Sa5] M. Saito,Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000),283–331.

[Sa6] M. Saito,Arithmetic mixed sheaves, Inv. Math. 144 (2001), 533–569.

[SS] L. Saper and M. Stern,L2-cohomology of arithmetic varieties, Ann. Math. (2)132:1 (1990), 1–69.

[SZ] L. Saper and S. Zucker,An introduction toL2-cohomology, in Several ComplexVariables and Complex Geometry(edited by E. Bedford et al.), Proc. Symp. PureMath., Vol. 52, Part 2, Amer. Math. Soc. (1991), 519–534.

[Si] P. H. Siegel,Witt spaces: A geometric cycle theory for KO-homology at oddprimes,Amer. J. Math. 105 (1983), 1067–1105.

[Sch1] J. Schurmann,Topology of singular spaces and constructible sheaves, Mono-grafie Matematyczne 63 (New Series), Birkhauser, Basel (2003).

[Sch2] J. Schurmann,A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes, preprint, arXiv:math.AG/0202175

[Sch3] J. Schurmann,Characteristic classes of mixed Hodge modules, in this volume.

[SY] J. Schurmann and S. Yokura,A survey of characteristic classes of singular spaces,in “Singularity Theory: Dedicated to Jean-Paul Brasselet on his 60th birthday”,edited by Denis Cheniot et al., World Scientific (2007), 865–952.


[Schw] M.-H. Schwartz,Classes caracteristiques definies par une stratification d’unevariete analytique complexe, C. R. Acad. Sci. Paris 260 (1965), 3262–3264 and3535–3537.

[Sh] J. L. Shaneson,Characteristic classes, lattice points, and Euler–MacLaurin for-mulae, Proc. Intern. Congress Math., Zurich 1994, vol. 1 (1995), 612–624

[Sr] V. Srinivas,The Hodge characteristic, Lectures inJet Schemes Seminar, MSRI,December 2002, Manuscript preprint (2002).

[Stong] R. E. Stong,Notes on Cobordism Theory, Princeton Univ. Press, 1968.

[Su] D. Sullivan,Combinatorial invariants of analytic spaces, Lect. Notes Math. 192(1970), 165–168.

[Ta] C. Taubes,S1 actions and elliptic genera, Comm. Math. Phys., 122 (1989), 455–526.

[Thom] R. Thom,Les classes caracteristiques de Pontrjagin des varietes triangulees,Symp. Intern. de Topologia Algebraica, Unesco (1958).

[To1] B. Totaro,Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc.10 (1997), 467–493.

[To2] B. Totaro,Chern numbers for singular varieties and elliptic homology, Ann.Math. 151 (2000), 757–791.

[To3] B. Totaro, Topology of singular algebraic varieties, Proc. Int. Cong. Math.Beijing, Vol. 1 (2002), 533–541.

[To4] B. Totaro, The elliptic genus of a singular variety, pp. 360–364 in [Mi-Ra](2007).

[Wei] S. Weinberger,The topological classification of stratified spaces, University ofChicago Press, Chicago, 1994.

[Wil] Floyd L. Williams, Lectures on zeta functions, L-functions and modular formswith some physical applications, in A window into zeta and modular physics(editedby Klaus Kirsten and Floyd L. Williams), MSRI Publications vol. 57, CambridgeUniv. Press, New York (2010), 7–100.

[Wit] E. Witten, The index of the Dirac operator in loop spaces, pp. 161–181 in [La].

[Wlo] J. Włodarczyk,Toroidal varieties and the weak factorization theorem, Inv. Math.154 (2003), 223–331.

[Y1] S. Yokura,On the universality of Baum-Fulton-MacPherson’s Riemann-Roch forsingular varieties, Proc. Japan. Acad. 68 (1992), 119–122.

[Y2] S. Yokura,A generalized Grothendieck-Riemann-Roch theorem for Hirzebruch’s�y-characteristic andTy-characteristic, Publ. Res. Inst. Math. Sci. 30 (1994), 603–610.

[Y2] S. Yokura,On Cappell-Shaneson’s homology L-class of singular algebraic vari-eties, Trans. Amer. Math. Soc. 347 (1995), 1005–1012.

[Y4] S. Yokura, A singular Riemann-Roch theorem for Hirzebruch characteristics.,Banach Center Publ. 44 (1998), 257–268.

418 SHOJI YOKURA

[Y5] S. Yokura, Constructible functions and Hodge polynomials, unpublished note,2003.

[Y6] S. Yokura,Characteristic classes of proalgebraic varieties and motivic measures,arXiv:math/0606352v2.

[Y7] S. Yokura,Oriented bivariant theory, I, International J. Math. 20:10 (2009), 1305–1334.

[Y8] S. Yokura,Motivic Milnor classes, J. Singularities, 1 (2010), 39–59.

[You] B. Youssin,Witt groups of derived categories, K-Theory 11 (1997), 373–395.

SHOJI YOKURA

DEPARTMENT OFMATHEMATICS AND COMPUTER SCIENCE

FACULTY OF SCIENCE

KAGOSHIMA UNIVERSITY

21-35 KORIMOTO 1-CHOME

KAGOSHIMA 890-0065JAPAN

[email protected]


Characteristic classes of mixed Hodge modulesJORG SCHURMANN

ABSTRACT. This paper is an extended version of an expository talk given atthe workshop “Topology of Stratified Spaces” at MSRI in September 2008.It gives an introduction and overview about recent developments on the in-teraction of the theories of characteristic classes and mixed Hodge theory forsingular spaces in the complex algebraic context.

It uses M. Saito’s deep theory of mixed Hodge modules as a black box,thinking about them as “constructible or perverse sheaves of Hodge struc-tures”, having the same functorial calculus of Grothendieck functors. Forthe “constant Hodge sheaf”, one gets the “motivic characteristic classes” ofBrasselet, Schurmann, and Yokura, whereas the classes of the “intersectionhomology Hodge sheaf” were studied by Cappell, Maxim, and Shaneson. Theclasses associated to “good” variation of mixed Hodge structures where studiedin connection with understanding the monodromy action by these three authorstogether with Libgober, and also by the author.

There are two versions of these characteristic classes. TheK-theoreticalclasses capture information about the graded pieces of the filtered de Rhamcomplex of the filteredD-module underlying a mixed Hodge module. Appli-cation of a suitable Todd class transformation then gives classes in homology.These classes are functorial for proper pushdown and exterior products, to-gether with some other properties one would expect for a satisfactory theoryof characteristic classes for singular spaces.

For “good” variation of mixed Hodge structures they have an explicit clas-sical description in terms of “logarithmic de Rham complexes”. On a pointspace they correspond to a specialization of the Hodge polynomial of a mixedHodge structure, which one gets by forgetting the weight filtration.

We also indicate some relations with other subjects of the conference, likeindex theorems, signature,L-classes, elliptic genera and motivic characteristicclasses for singular spaces.

419

420 JORG SCHURMANN

CONTENTS

1. Introduction 4202. Hodge structures and genera 4233. Characteristic classes of variations of mixed Hodge structures 4304. Calculus of mixed Hodge modules 4405. Characteristic classes of mixed Hodge modules 450Acknowledgements 467References 467

1. Introduction

This paper gives an introduction and overview about recent developments onthe interaction of the theories of characteristic classes and mixed Hodge theoryfor singular spaces in the complex algebraic context. The reader is not assumedto have a background on any of these subjects, and the paper can also be usedas a bridge for communication between researchers on one of these subjects.

General references for the theory of characteristic classes of singular spacesare the survey [48] and the paper [55] in these proceedings. As references formixed Hodge theory one can use [38; 52], as well as the nice paper [37] forexplaining the motivic viewpoint to mixed Hodge theory. Finally as an intro-duction to M. Saito’s deep theory of mixed Hodge modules one can use [38,Chapter 14], [41] as well as the introduction [45].

The theory of mixed Hodge modules is used here more or less as ablack box;we think about them as constructible or perverse sheaves of Hodge structures,having the same functorial calculus of Grothendieck functors. The underlyingtheory of constructible and perverse sheaves can be found in[7; 30; 47].

For the “constant Hodge sheaf”QHZ one gets the “motivic characteristic

classes” of Brasselet, Schurmann, and Yokura [9], as explained in these pro-ceedings [55]. The classes of the “intersection homology Hodge sheaf”IC H

Z

were studied by Cappell, Maxim, and Shaneson in [10; 11]. Also, the classesassociated to “good” variation of mixed Hodge structures where studied viaAtiyah–Meyer type formulae by Cappell, Libgober, Maxim, and Shaneson in[12; 13]. For a summary compare also with [35].

There are two versions of these characteristic classes, themotivic Chernclass transformation MHCy and themotivic Hirzebruch class transformationMHTy�. TheK-theoretical classesMHCy capture information about the gradedpieces of the filtered de Rham complex of the filteredD-module underlying amixed Hodge module. Application of a suitable twistingtd.1Cy/ of the Toddclass transformationtd� of Baum, Fulton, and MacPherson [5; 22] then givesthe classesMHTy�D td.1Cy/ıMHCy in homology. It is themotivic Hirzebruchclass transformation MHTy�, which unifies three concepts:

CHARACTERISTIC CLASSES OF MIXED HODGE MODULES 421

.yD�1/ the (rationalized)Chern class transformationc� of MacPherson [34];

.y D 0/ theTodd class transformationtd� of Baum–Fulton–MacPherson [5];

.y D 1/ theL-class transformationL� of Cappell and Shaneson [14].

(Compare with [9; 48] and also with [55] in these proceedings.) But in thispaper we focus on theK-theoretical classesMHCy , because these imply thenalso the corresponding results forMHTy� just by application of the (twisted)Todd class transformation. So themotivic Chern class transformation MHCy

studied here is really the basic one!

Here we explain the functorial calculus of these classes, first stating in a veryprecise form the key results used from Saito’s theory of mixed Hodge modules,and then explaining how to get from this the basic results about the motivicChern class transformationMHCy . These results are illustrated by many inter-esting examples. For the convenience of the reader, the mostgeneral resultsare only stated near the end of the paper. In fact, while most of the paper is adetailed survey of theK-theoretical version of the theory as developed in [9; 12;13; 35], it is this last section that contains new results on the important functorialproperties of these characteristic classes. The first two sections do not use mixedHodge modules and are formulated in the now classical language of (variationof) mixed Hodge structures. Here is the plan of the paper:

SECTION 2 introduces pure and mixed Hodge structures and the correspondingHodge genera, such as theE-polynomial and the�y-genus. These are suit-able generating functions of Hodge numbers with�y using only the Hodgefiltration F , whereas theE-polynomial also uses the weight filtration. Wealso carefully explain why only the�y-genus can be further generalized tocharacteristic classes, i.e., why one has to forget the weight filtration for ap-plications to characteristic classes.

SECTION 3 motivates and explains the notion of a variation (or family) of pureand mixed Hodge structures over a smooth (or maybe singular)base. Basicexamples come from the cohomology of the fibers of a family of complex al-gebraic varieties. We also introduce the notion of a “good” variation of mixedHodge structures on a complex algebraic manifoldM , to shorten the notionfor a graded polarizable variation of mixed Hodge structures onM that isad-missiblein the sense of Steenbrink and Zucker [50] and Kashiwara [28], withquasi-unipotent monodromyat infinity, i.e., with respect to a compactificationM of M by a compact complex algebraic manifoldM , with complementD WDM nM a normal crossing divisor with smooth irreducible components.Later on these will give the basic example of so-called “smooth” mixed Hodgemodules. And for these good variations we introduce a simplecohomologicalcharacteristic class transformationMHCy , which behaves nicely with respect

422 JORG SCHURMANN

to smooth pullback, duality and (exterior) products. As a first approximationto more general mixed Hodge modules and their characteristic classes, wealso study in detail functorial properties of the canonicalDeligne extensionacross a normal crossing divisorD at infinity (as above), leading tocohomo-logical characteristic classesMHCy.j�. � // defined in terms of “logarithmicde Rham complexes”. These classes of good variations have been studiedin detail in [12; 13; 35], and most results described here arenew functorialreformulations of the results from these sources.

SECTION 4 starts with an introduction to Saito’s functorial theory of algebraicmixed Hodge modules, explaining its power in many examples,includinghow to get a pure Hodge structure on the global intersection cohomologyIH �.Z/ of a compact complex algebraic varietyZ. From this we deducethe basic calculus of Grothendieck groupsK0.MHM. � // of mixed Hodgemodules needed for our motivic Chern class transformationMHCy . We alsoexplain the relation to the motivic viewpoint coming from relative Grothen-dieck groups of complex algebraic varieties.

SECTION 5.1 is devoted to the definition of our motivic characteristic homologyclass transformationsMHCy and MHTy� for mixed Hodge modules. BySaito’s theory they commute with push down for proper morphisms, and ona compact space one gets back the corresponding�y-genus by pushing downto a point, i.e., by taking the degree of these characteristic homology classes.

SECTIONS5.2 AND 5.3 finally explain other important functoriality properties:

(1) multiplicativity for exterior products;

(2) the behavior under smooth pullback given by a Verdier Riemann–Rochformula;

(3) a “going up and down” formula for proper smooth morphisms;

(4) multiplicativity betweenMHCy andMHCy for a suitable (co)homologicalpairing in the context of a morphism with smooth target (as special casesone gets interesting Atiyah and Atiyah–Meyer type formulae, as studied in[12; 13; 35]);

(5) the relation betweenMHCy and duality, i.e., the Grothendieck dualitytransformation for coherent sheaves and Verdier duality for mixed Hodgemodules;

(6) the identification ofMHT�1� with the (rationalized) Chern class trans-formation c� ˝ Q of MacPherson for the underlying constructible sheafcomplex or function.

Note that such a functorial calculus is expected for any goodtheory of functorialcharacteristic classes of singular spaces (compare [9; 48]):


� for MacPherson’s Chern class transformationc� compare with [9; 31; 34;48];

� for the Baum–Fulton–MacPherson Todd class transformationtd� comparewith [5; 6; 9; 22; 24; 48];

� for Cappell and Shaneson’sL-class transformationL� compare with [2; 3;4; 9; 14; 48; 49; 54].

The counterpart of mixed Hodge modules in these theories areconstructiblefunctions and sheaves (forc�), coherent sheaves (fortd�) and selfdual perverseor constructible sheaf complexes (forL�). The cohomological counterpart of thesmooth mixed Hodge modules (i.e., good variation of mixed Hodge structures)are locally constant functions and sheaves (forc�), locally free coherent sheavesor vector bundles (for the Chern characterch�) and selfdual local systems (fora twisted Chern character of theKO-classes of Meyer [36]).

In this paper we concentrate mainly on pointing out the relation and analogyto theL-class story related to important signature invariants, because these arethe subject of many other talks from the conference given in more topologicalterms. Finally also some relations to other themes of the conference, like indextheorems,L2-cohomology, elliptic genera and motivic characteristic classes forsingular spaces, will be indicated.

2. Hodge structures and genera

2A. Pure Hodge structures. Let M be a compactKahler manifold(e.g., acomplex projective manifold) of complex dimensionm. By classical Hodgetheory one gets the decomposition (for0� n� 2m)

H n.M;C/DM

pCqDn

H p;q.M / (2-1)

of the complex cohomology ofM into the spacesH p;q.M / of harmonic formsof type .p; q/. This decomposition doesn’t depend on the choice of a Kahlerform (or metric) onM , and for a complex algebraic manifoldM it is of alge-braic nature. Here it is more natural to work with theHodge filtration

F i.M / WDM

p�i

H p;q.M / (2-2)

so thatH p;q.M /DFp.M /\Fq.M /, with Fq.M / the complex conjugate ofFq.M / with respect to the real structureH n.M;C/DH n.M;R/˝C. If

˝�

MD ŒO

M

d��! � � �

d��! ˝m

M�

424 JORG SCHURMANN

denotes the usual holomorphic de Rham complex (withOM in degree zero),then one gets

H �.M;C/DH �.M; ˝�

M /

by the holomorphic Poincare lemma, and the Hodge filtration is induced fromthe “stupid” decreasing filtration

Fp˝�

MD Œ0 ��! � � � 0 ��! ˝

pM

d��! � � �

d��! ˝m

M�: (2-3)

More precisely, the correspondingHodge to de Rham spectral sequencedegen-erates atE1, with

Ep;q1DH q.M; ˝

pM/'H p;q.M /: (2-4)

The same results are true for a compact complex manifoldM that is onlybimeromorphic to a Kahler manifold(compare [38, Corollary 2.30], for ex-ample). This is especially true for a compact complex algebraic manifoldM .Moreover in this case one can calculate by Serre’s GAGA theoremH �.M; ˝�

M/

also with the algebraic (filtered) de Rham complex in the Zariski topology.

Abstracting these properties, one can say theH n.M;Q/ gets an inducedpureHodge structure of weightn in the following sense:

DEFINITION 2.1. LetV be a finite-dimensional rational vector space. A (ratio-nal) Hodge structure of weightn on V is a decomposition

VC WD V ˝Q C DM

pCqDn

V p;q; with V q;p D V p;q (Hodge decomposition).

In terms of the (decreasing)Hodge filtrationF iVC WDL

p�i V p;q, this is equiv-alent to the condition

FpV \FqV D f0g wheneverpC q D nC 1 (n-opposed filtration).

ThenV p;q D Fp \Fq, with hp;q.V / WD dim.V p;q/ the correspondingHodgenumber.

If V;V 0 are rational vector spaces with Hodge structures of weightn andm,then V ˝ V 0 gets an induced Hodge structure of weightnCm, with Hodgefiltration

Fk.V ˝V 0/C WDM

iCjDk

F iVC ˝F jV 0

C : (2-5)

Similarly the dual vector spaceV _ gets an induced Hodge structure of weight�n, with

Fk.V _

C / WD .F�kVC/

_: (2-6)


A basic example is theTate Hodge structureof weight�2n 2 Z given by theone-dimensional rational vector space

Q.n/ WD .2� i/n �Q � C; with Q.n/C D .Q.n/C/�n;�n.

Then integration defines an isomorphism

H 2.P 1.C/;Q/'Q.�1/;

with Q.�n/DQ.�1/˝n, Q.1/DQ.�1/_ andQ.n/DQ.1/˝n for n> 0.

DEFINITION 2.2. Apolarizationof a rational Hodge structureV of weightn isa rational.�1/n-symmetric bilinear formS on V such that

S.Fp;Fn�pC1/D 0 for all p

and

ip�qS.u; Nu/ > 0 for all nonzerou 2 V p;q :

So forn evenone gets in particular

.�1/p�n=2S.u; Nu/ > 0 for all q and all nonzerou 2 V p;q : (2-7)

V is calledpolarizableif such a polarization exists.

For example, the cohomologyH n.M;Q/ of a projective manifold is polarizableby the choice of a suitable Kahler form! Also note that a polarization of a rationalHodge structureV of weightn induces an isomorphism of Hodge structures (ofweightn):

V ' V _.�n/ WD V _˝Q Q.�n/:

So if we choose the isomorphism of rational vector spaces

Q.�n/D .2� i/�n �Q 'Q;

then a polarization induces a.�1/n-symmetric duality isomorphismV ' V _.

2B. Mixed Hodge structures. The cohomology (with compact support) ofa singular or noncompact complex algebraic variety, denoted by H n

.c/.X;Q/,

can’t have a pure Hodge structure in general, but by Deligne’s work [20; 21] itcarries a canonical functorial (graded polarizable)mixed Hodge structurein thefollowing sense:

DEFINITION 2.3. A finite-dimensional rational vector spaceV has a mixedHodge structure if there is a (finite) increasingweight filtrationW DW� on V

(by rational subvector spaces), and a (finite) decreasing Hodge filtrationF DF �

on VC , such thatF induces a Hodge structure of weightn on GrWn V WD

WnV =Wn�1V for all n. Such a mixed Hodge structure is called (graded) po-larizable if each graded pieceGrW

n V is polarizable.

426 JORG SCHURMANN

A morphism of mixed Hodge structures is just a homomorphism of rationalvector spaces compatible with both filtrations. Such a morphism is thenstrictlycompatible with both filtrations, so that the categorymHs.p/ of (graded po-larizable) mixed Hodge structures is an abelian category, with GrW

�;Gr�

F andGr�

F GrW�

preserving short exact sequences. The categorymHs.p/ is also en-dowed with a tensor product and a duality. � /_, where the correspondingHodge and weight filtrations are defined as in (2-5) and (2-6).So for a complexalgebraic varietyX one can consider its cohomology class

ŒH �

.c/.X /� WDX

i

.�1/i � ŒH i.c/.X;Q/� 2K0.mHs.p//

in the Grothendieck groupK0.mHs.p// of (graded polarizable) mixed Hodgestructures. The functoriality of Deligne’s mixed Hodge structure means, inparticular, that for a closed complex algebraic subvarietyY � X , with opencomplementU DXnY , the corresponding long exact cohomology sequence

� � �H ic .U;Q/!H i

c .X;Q/!H ic .Y;Q/! � � � (2-8)

is an exact sequence of mixed Hodge structures. Similarly, for complex alge-braic varietiesX;Z, the Kunneth isomorphism

H �

c .X;Q/˝H �

c .Z;Q/'H �

c .X �Z;Q/ (2-9)

is an isomorphism of mixed Hodge structures. Let us denote byK0.var=pt/ theGrothendieck group of complex algebraic varieties, i.e., the free abelian group ofisomorphism classesŒX � of such varieties divided out by theadditivity relation

ŒX �D ŒY �C ŒXnY �

for Y � X a closed complex subvariety. This is then a commutative ringwithaddition resp. multiplication induced by the disjoint union resp. the product ofvarieties. So by (2-8) and (2-9) we get an induced ring homomorphism

�Hdg WK0.var=pt/!K0.mHs.p//I ŒX �‘ ŒH �

c .X /�: (2-10)

2C. Hodge genera. TheE-polynomial

E.V / WDX

p;q

hp;q.V / �upvq 2 ZŒu˙1; v˙1� (2-11)

of a rational mixed Hodge structureV with Hodge numbers

hp;q.V / WD dimC GrpF

GrWpCq.VC/;

induces aring homomorphism

E WK0.mHs.p//! ZŒu˙1; v˙1�; with E.Q.�1//D uv.


Note thatE.V /.u; v/ is symmetricin u andv, sinceh.V /DP

n h.WnV / andV q;p D V p;q for a pure Hodge structure. With respect toduality one has inaddition the relation

E.V _/.u; v/DE.V /.u�1; v�1/: (2-12)

Later on we will be mainly interested in the specialized ringhomomorphism

�y WDE.�y; 1/ WK0.mHs.p//! ZŒy˙1�; with �y.Q.�1//D�y,

defined by

�y.V / WDX

p

dimC.GrpF.VC// � .�y/p: (2-13)

So here one uses only the Hodge and forgets the weight filtration of a mixedHodge structure. With respect toduality one has then the relation

�y.V_/D �1=y.V /: (2-14)

Note that��1.V /D dim.V / and for a pure polarized Hodge structureV ofweightn one has by�1.V /D .�1/n�1.V

_/D .�1/n�1.V / and (2-7):

�1.V /D

�

0 for n odd,sgnV for n even,

where sgnV is thesignatureof the induced symmetric bilinear form.�1/n=2S

on V . A similar but deeper result is the famousHodge index theorem(compare[52, Theorem 6.3.3], for example):

�1.ŒH�.M /�/D sgn.H m.M;Q//

for M a compact Kahler manifold of even complex dimensionm D 2n. Herethe right side denotes the signature of the symmetric intersection pairing

H m.M;Q/�H m.M;Q/[

��! H 2m.M;Q/'Q:

The advantage of�y compared toE (and the use of�y in the definition)comes from the following question:

Let E.X / WD E.ŒH �.X /�/ for X a complex algebraic variety. For M acompact complex algebraic manifold one gets by(2-4):

E.M /DX

p;q�0

.�1/pCq �dimC H q.M; ˝pM/ �upvq:

Is there a(normalized multiplicative) characteristic class

cl� W Iso.C �VB.M //!H �.M /Œu˙1; v˙1�

428 JORG SCHURMANN

of complex vector bundles such that the E-polynomial is a characteristic numberin the sense that

E.M /D ].M / WD deg.cl�.TM /\ ŒM �/ 2H �.pt/Œu˙1; v˙1� (2-15)

for any compact complex algebraic manifoldM with fundamental classŒM �?So the cohomology classcl�.V / 2 H �.M /Œu˙1; v˙1� should only depend

on the isomorphism class of the complex vector bundleV overM and commutewith pullback. Multiplicativity says

cl�.V /D cl�.V 0/[ cl�.V 00/ 2H �.M /Œu˙1; v˙1�

for any short exact sequence0!V 0!V !V 00! 0 of complex vector bundleson M . Finally cl� is normalized ifcl�.trivial/ D 1 2 H �.M / for any trivialvector bundle. Then the answer to the question isnegative, because there areunramified coveringsp W M 0 ! M of elliptic curvesM;M 0 of (any) degreed > 0. Thenp�TM ' TM 0 andp�.ŒM

0�/D d � ŒM �, so the projection formulawould give for the topological characteristic numbers the relation

].M 0/D d � ].M /:

But one has

E.M /D .1�u/.1� v/DE.M 0/¤ 0;

so the equalityE.M / D ].M / is not possible! Here we don’t need to askcl� to be multiplicative or normalized. But if we use the invariant �y.X / WD

�y.ŒH�.X /�/, then�y.M /D0 for an elliptic curve, and�y.M / is a characteris-

tic number in the sense above by the famousgeneralized Hirzebruch Riemann–Roch theorem[27]:

THEOREM 2.4 (GHRR). There is a unique normalized multiplicative charac-teristic class

T �

y W Iso.C �VB.M //!H �.M;Q/Œy�

such that

�y.M /D deg.T �

y .TM /\ ŒM �/D hT �

y .TM /; ŒM �i 2 ZŒy��QŒy�

for any compact complex algebraic manifoldM . Here h � ; � i is the Kroneckerpairing between cohomology and homology.

TheHirzebruch classT �y and�y-genus unify the following (total) characteristic

classes and numbers:y T �

y [name] �y [name]

�1 c� Chern class � Euler characteristic0 td� Todd class �a arithmetic genus1 L� L-class sgn signature


In fact, gHRR is just a cohomological version of the following K-theoreticalcalculation. LetM be a compact complex algebraic manifold, so that

�y.M /DX

p;q�0

.�1/pCq �dimC H q.M; ˝pM/ � .�y/p

DX

p�0

�.H �.M; ˝pM// �yp: (2-16)

Let us denote byK0an.Y / (or Gan

0.Y /) the Grothendieck group of the exact

(or abelian) category of holomorphic vector bundles (or coherentOY -modulesheaves) on the complex varietyY , i.e., the free abelian group of isomorphismclassesV of such vector bundles (or sheaves), divided out by the relation

ŒV �D ŒV 0�C ŒV 00� for any short exact sequence0! V 0! V ! V 00! 0.

ThenGan0.Y / (or K0

an.Y /) is of (co)homological nature, with

f� WGan0 .X /!Gan

0 .Y /; ŒF �‘X

i�0

.�1/i ŒRif�F �

the functorial pushdown for a proper holomorphic mapf WX!Y . In particular,for X compact, the constant mapk WX ! pt is proper, with

�.H �.X;F//D k�.ŒF �/ 2Gan0 .pt/'K0

an.pt/' Z:

Moreover, the tensor productOYinduces a natural pairing

\D˝ WK0an.Y /�Gan

0 .Y /!Gan0 .Y /;

where we identify a holomorphic vector bundleV with its locally free coherentsheaf of sectionsV . So forX compact we can define aKronecker pairing

K0an.X /�Gan

0 .X /! Gan0 .pt/' ZI hŒV �; ŒF �i WD k�.ŒV˝OX

F �/:

The total �-classof the dual vector bundle

�y.V_/ WD

X

i�0

�i.V _/ �yi

defines a multiplicative characteristic class

�y.. � /_/ WK0

an.Y /!K0an.Y /Œy�:

And for a compact complex algebraic manifoldM one gets the equality

�y.M /DX

i�0

k�Œ˝iM � �yi

D h�y.T�M /; ŒOM �i 2Gan

0 .pt/Œy�' ZŒy�: (2-17)

430 JORG SCHURMANN

3. Characteristic classes of variations of mixed Hodge structures

This section explains the definition ofcohomologicalcharacteristic classesassociated to good variations of mixed Hodge structures on complex algebraicand analytic manifolds. These were previously considered in [12; 13; 35] inconnection with Atiyah–Meyer type formulae of Hodge-theoretic nature. Herewe also consider important functorial properties of these classes.

3A. Variation of Hodge structures. Let f W X ! Y be aproper smoothmorphism of complex algebraic varieties or aprojective smoothmorphism ofcomplex analytic varieties. Then the higher direct image sheaf L D Ln WD

Rnf�QX is a locally constant sheafon Y with finite-dimensional stalks

Ly D .Rnf�QX /y DH n.ff D yg;Q/

for y 2 Y . Let L WD L˝QYOY ' Rnf�.˝

�

X=Y/ denote the corresponding

holomorphic vector bundle (or locally free sheaf), with�X=Y

therelative holo-morphic de Rham complex. Then the stupid filtration of �

X=Ydetermines a

decreasing filtrationF of L by holomorphic subbundlesFpL, with

GrpF..RpCqf�QX /˝QY

OY /'Rqf�.˝p

X=Y/; (3-1)

inducing for ally 2 Y the Hodge filtrationF on the cohomology

H n.ff D yg;Q/˝C ' Ljy

of the compact and smooth algebraic fiberff D yg (compare [38, Chapter 10]).If Y (and therefore alsoX ) is smooth, thenL gets an inducedintegrable Gauss–Manin connection

r W L! L˝OY˝1

Y ; with L' kerr andr ır D 0;

satisfying theGriffiths transversality condition

r.FpL/� Fp�1

L˝OY˝1

Y for all p. (3-2)

This motivates the following notion:

DEFINITION 3.1. A holomorphic family.L;F / of Hodge structures of weightn on the reduced complex spaceY is a local systemL with rational coefficientsand finite-dimensional stalks onY , and a decreasing filtrationF of LDL˝QY

OY by holomorphic subbbundlesFpL such thatF determines byLy˝Q C '

Ljy a pure Hodge structure of weightn on each stalkLy (y 2 Y ).If Y is a smooth complex manifold, then such a holomorphic family.L;F /

is called avariation of Hodge structures of weightn if, in addition, Griffithstransversality (3-2) holds for the induced connectionr W L! L˝OY

˝1Y

.


Finally apolarizationof .L;F / is a pairing of local systemsS WL˝QYL!

QY that induces a polarization of Hodge structures on each stalk Ly (y 2 Y ).

For example in the geometric case above, one can get such a polarization onLDRnf�QX for f W X ! Y a projective smoothmorphism of complex algebraic(or analytic) varieties. The existence of a polarization isneeded for example forthe following important result of Schmid [46, Theorem 7.22]:

THEOREM 3.2 (RIGIDITY ). Let Y be a connected complex manifold Zariskyopen in a compact complex analytic manifoldY , with .L;F / a polarizablevariation of pure Hodge structures onY . ThenH 0.Y;L/ gets an induced Hodgestructure such that the evaluation mapH 0.Y;L/! Ly is an isomorphism ofHodge structures for ally 2 Y . In particular the variation.L;F / is constant ifthe underlying local systemL is constant.

3B. Variation of mixed Hodge structures. If one considers a morphismf WX ! Y of complex algebraic varieties withY smooth, which is a topologicalfibration with possible singular or noncompact fiber, then the locally constantdirect image sheavesLDLn WDRnf�QX (n�0) arevariations of mixed Hodgestructuresin the sense of the following definitions.

DEFINITION 3.3. LetY be a reduced complex analytic space. Aholomorphicfamily of mixed Hodge structureson Y consists of

(1) a local systemL of rational vector spaces onY with finite-dimensionalstalks,

(2) a finite decreasingHodge filtrationF of L D L˝QYOY by holomorphic

subbundlesFpL,(3) a finite increasingweight filtrationW of L by local subsystemsWnL,

such that the induced filtrations onLy'Ly˝Q C andLy define a mixed Hodgestructure on all stalksLy (y 2 Y ).

If Y is a smooth complex manifold, such a holomorphic family.L;F;W / iscalled avariation of mixed Hodge structuresif, in addition, Griffiths transver-sality (3-2) holds for the induced connectionr W L! L˝OY

˝1Y

.Finally, .L;F;W / is calledgraded polarizableif the induced family (or vari-

ation) of pure Hodge structuresGrWn L (with the induced Hodge filtrationF ) is

polarizable for alln.

With the obvious notion of morphisms, the two categoriesFmHs.p/.Y / andVmHs.p/.Y / of (graded polarizable) families and variations of mixed Hodgestructures onY become abelian categories with a tensor product˝ and duality. � /_. Again, any such morphism is strictly compatible with the Hodge andweight filtrations. Moreover, one has for a holomorphic mapf W X ! Y (of

432 JORG SCHURMANN

complex manifolds) a functorial pullback

f � W FmHs.p/.Y /! FmHs.p/.X / (or f � W VmHs.p/.Y /! VmHs.p/.X /),

commuting with tensor product and duality. � /_. On a point spacept onejust gets back the category

FmHs.p/.pt/D VmHs.p/.pt/DmHs.p/

of (graded polarizable) mixed Hodge structures. Using the pullback under theconstant mapk W Y ! pt , we get the constant family (or variation) of TateHodge structuresQY .n/ WD k�Q.n/ on Y .

3C. Cohomological characteristic classes. The Grothendieck groupK0an.Y /

of holomorphic vector bundles on the complex varietyY is a commutative ringwith multiplication induced by and has a duality involution induced by. � /_.For a holomorphic mapf W X ! Y one has a functorial pullbackf � of ringswith involutions. The situation is similar forK0

an.Y /Œy˙1�, if we extend the

duality involution by

.ŒV � �yk/_ WD ŒV _� � .1=y/k :

For a family (or variation) of mixed Hodge structures.L;F;W / on Y let usintroduce the characteristic class

MHCy..L;F;W // WDX

p

ŒGrpF.L/� � .�y/p 2K0

an.Y /Œy˙1�: (3-3)

Because morphisms of families (or variations) of mixed Hodge structures arestrictly compatible with the Hodge filtrations, we get induced group homomor-phisms of Grothendieck groups:

MHCy WK0.FmHs.p/.Y //!K0an.Y /Œy

˙1�;

MHCy WK0.VmHs.p/.Y //!K0an.Y /Œy

˙1�:

Note thatMHC�1..L;F;W // D ŒL� 2 K0an.Y / is just the class of the asso-

ciated holomorphic vector bundle. And forY D pt a point, we get back the�y-genus:

�y DMHCy WK0.mHs.p//DK0.FmHs.p/.pt//!K0an.pt/Œy˙1�D ZŒy˙1�:

THEOREM 3.4. The transformations

MHCy WK0.FmHs.p/.Y //!K0an.Y /Œy

˙1�;

MHCy WK0.VmHs.p/.Y //!K0an.Y /Œy

˙1�;


are contravariant functorial, and are transformations of commutative rings withunit, i.e., they commute with products and respect units: MHCy.ŒQY .0/�/ D

ŒOY �. Similarly they respect duality involutions:

MHCy.Œ.L;F;W /_�/DX

p

Œ.Gr�pF.L//_� �.�y/pD

�

MHCy.Œ.L;F;W /�/�_:

EXAMPLE 3.5. Letf W X ! Y be aproper smoothmorphism of complexalgebraic varieties or aprojective smoothmorphism of complex analytic va-rieties, so that the higher direct image sheafLn WD Rnf�QX (n � 0) with theinduced Hodge filtration as in (3-1) defines a holomorphic family of pure Hodgestructures onY . If m is the complex dimension of the fibers, thenLn D 0 forn> 2m, so one can define

ŒRf�QX � WD

2mX

nD0

.�1/n � Œ.Rnf�QX ;F /� 2K0.FmHs.Y //:

Then one gets, by (3-1),

MHCy.ŒRf�QX �/DX

p;q�0

.�1/pCq � ŒRqf�˝p

X=Y� � .�y/p

DX

p�0

f�Œ˝p

X=Y� �yp

DW f�

�

�y.T�

X=Y /�

2K0an.Y /Œy�: (3-4)

Assume moreover that

(a) Y is a connected complex manifold Zarisky open in a compact complexanalytic manifoldY , and

(b) all direct images sheavesLn WDRnf�QX (n� 0) areconstant.

Then one gets by therigidity theorem3.2 (for z 2 Y ):

f�

�

�y.T�

X=Y /�

D �y.ff D zg/ � ŒOY � 2K0an.Y /Œy�:

COROLLARY 3.6 (MULTIPLICATIVITY ). Letf WX!Y be a smooth morphismof compact complex algebraic manifolds, with Y connected. Let T �

X=Ybe the

relative holomorphic cotangent bundle of the fibers, fitting into the short exactsequence

0! f �T �Y ! T �X ! T �

X=Y ! 0:

Assume all direct images sheavesLn WD Rnf�QX (n � 0) are constant, i.e.,�1.Y / acts trivially on the cohomologyH �.ff D zg/ of the fiber. Then one

434 JORG SCHURMANN

gets the multiplicativity of the�y-genus(with k W Y ! pt the constant map):

�y.X /D .k ıf /�Œ�y.T�X /�

D k�f�

�

Œ�y.T�

X=Y /�˝ f�Œ�y.T

�Y /��

D k�

�

�y.ff D zg/ � Œ�y.T�Y /�

�

D �y.ff D zg/ ��y.Y /: (3-5)

REMARK 3.7. The multiplicativity relation (3-5) specializes fory D 1 to theclassical multiplicativity formula

sgn.X /D sgn.ff D zg/ � sgn.Y /

of Chern, Hirzebruch, and Serre [16] for the signature of an oriented fibrationof smooth coherently oriented compact manifolds, if�1.Y / acts trivially on thecohomologyH �.ff D zg/ of the fiber. So it is a Hodge theoretic counterpartof this. Moreover, the corresponding Euler characteristicformula fory D�1

�.X /D �.ff D zg/ ��.Y /

is even truewithout�1.Y / acting trivially on the cohomologyH �.ff D zg/ ofthe fiber!

The Chern–Hirzebruch–Serre signature formula was motivational for manysubsequent works which studied monodromy contributions toinvariants (generaand characteristic classes). See, for exmaple, [1; 4; 10; 11; 12; 13; 14; 35; 36].

Instead of working with holomorphic vector bundles, we can of course alsouse only the underlying topological complex vector bundles, which gives theforgetful transformation

For WK0an.Y /!K0

top.Y /:

Here the target can also be viewed as the even part ofZ2-graded topologicalcomplex K-cohomology. Of course, the forgetful transformation is contra-variant functorial and commutes with productand with duality. � /_. Thisduality induces aZ2-grading onK0

top.Y /�

12

�

by splitting into the (anti-)invariantpart, and similarly forK0

an.Y /�

12

�

. Then the (anti-)invariant part ofK0top.Y /

�

12

�

can be identified with the even part ofZ4-graded topological realK-theoryKO0

top.Y /�

12

�

(andKO2top.Y /

�

12

�

).

Assume now that.L;F / is a holomorphic family of pure Hodge structures ofweight n on the complex varietyY , with a polarizationS W L˝QY

L! QY .This induces an isomorphism of families of pure Hodge structures of weightn:

L'L_.�n/ WDL_˝QY .�n/:


So if we choose the isomorphism of rational local systems

QY .�n/D .2� i/�n �QY 'QY ;

the polarization induces a.�1/n-symmetric duality isomorphismL ' L_ ofthe underlying local systems. And for such an (anti)symmetric selfdual localsystemL Meyer [36] has introduced aKO-characteristic class

ŒL�KO 2KO0top.Y /

�

12

�

˚KO2top.Y /

�

12

�

/DK0top.Y /

�

12

�

;

so that forY a compact oriented manifold of even real dimension2m the fol-lowing twisted signature formulais true:

sgn.H m.Y;L//D hch�.2.ŒL�KO//;L�.TM /\ ŒM �i: (3-6)

HereH m.Y;L/ gets an induced (anti)symmetric duality, with sgn.H m.Y;L//

defined as 0 in case of an antisymmetric pairing. Moreoverch� is the Cherncharacter,2 the second Adams operation andL� is the Hirzebruch–ThomL-class.

We now explain thatŒL�KO agrees up to some universal signs with

For.MHC1..L;F //:

The underlying topological complex vector bundle ofL has a natural real struc-ture, so that, as a topological complex vector bundle, one gets an orthogonaldecomposition

LDM

pCqDn

Hp;q; with Hp;q D FpL\FqLDHq;p,

withFor.MHC1..L;F //D

X

p evenq

ŒHp;q ��X

p oddq

ŒHp;q �: (3-7)

If n is even, both sums on the right are invariant under conjugation. And, by(2-7), .�1/�n=2 �S is positive definite on the corresponding real vector bundle�L

p even;q Hp;q�

R, and negative definite on

�L

p odd;q Hp;q�

R. So if we choose

the pairing.�1/n=2 �S for the isomorphismL' L_, then this agrees with thesplitting introduced by Meyer [36] in the definition of hisKO-characteristicclassŒL�KO associated to thissymmetricduality isomorphism ofL:

For.MHC1..L;F //D ŒL�KO 2KO0top.Y /

�

12

�

:

Similarly, if n is odd, both sums of the right hand side in (3-7) are exchangedunder conjugation. If we choose the pairing.�1/.nC1/=2 �S for the isomorphismL ' L_, then this agrees by Definition 2.2 with the splitting introduced by

436 JORG SCHURMANN

Meyer [36] in the definition of hisKO-characteristic classŒL�KO associated tothis antisymmetricduality isomorphism ofL:


�

12

�

:

COROLLARY 3.8. Let.L;F / be a holomorphic family of pure Hodge structuresof weightn on the complex varietyY , with a polarizationS chosen. The classŒL�KO introduced in[36] for the duality isomorphism coming from the pairing.�1/n.nC1/=2 �S is equal to


�

12

�

˚KO2top.Y /

�

12

�

DK0top.Y /

�

12

�

:

It is therefore independent of the choice of the polarization S . Moreover, thisidentification is functorial under pullback and compatiblewith products(as de-fined in[36, p. 26]for (anti)symmetric selfdual local systems).

There are Hodge theoretic counterparts of the twisted signature formula (3-6).Here we formulate a correspondingK-theoretical result. Let.L;F;W / be avariation of mixed Hodge structures on them-dimensional complex manifoldM . Then

H n.M;L/'H n.M;DR.L//

gets an induced (decreasing)F filtration coming from the filtration of the holo-morphic de Rham complex of the vector bundleL with its integrable connec-tion r:

DR.L/D ŒLr

��! � � �r

��! L˝OM˝m

M�

(with L in degree zero), defined by

FpDR.L/D ŒFpLr

��! � � �r

��! Fp�mL˝OM˝m

M�: (3-8)

Note that here we are using Griffiths transversality (3-2)!

The following result is due to Deligne and Zucker [56, Theorem 2.9, Lemma2.11] in the case of a compact Kahler manifold, whereas the case of a compactcomplex algebraic manifold follows from Saito’s general results as explained inthe next section.

THEOREM3.9. AssumeM is a compact Kahler manifold or a compact complexalgebraic manifold, with .L;F;W / a graded polarizable variation of mixed(or pure) Hodge structures onM . ThenH n.M;L/'H n.M;DR.L// gets aninduced mixed(or pure) Hodge structure withF the Hodge filtration. Moreover,the corresponding Hodge to de Rham spectral sequence degenerates atE1 sothat

GrpF.H n.M;L//'H n.M;Grp

FDR.L// for all n;p.


Therefore one gets as a corollary (compare [12; 13; 35]):

�y.H�.M;L//D

X

n;p

.�1/n �dimC

�

H n.M;GrpF

DR.L//�

� .�y/p

DX

p

��

H �.M;GrpF

DR.L//�

� .�y/p

DX

p;i

.�1/i ��

H �.M;Grp�iF

.L/˝OM˝i

M /�

� .�y/p

D k�

�

MHCy.L/˝�y.T�M /

�

DW hMHCy.L/; �y.T�M /\ ŒOM �i 2 ZŒy˙1�: (3-9)

3D. Good variation of mixed Hodge structures.

DEFINITION 3.10 (GOOD VARIATION). Let M be a complex algebraic mani-fold. A graded polarizable variation of mixed Hodge structures.L;F;W / onM

is called good if it isadmissiblein the sense of Steenbrink and Zucker [50] andKashiwara [28], withquasi-unipotent monodromyat infinity, i.e., with respect toa compactificationM of M by a compact complex algebraic manifoldM , withcomplementD WD M nM a normal crossing divisor with smooth irreduciblecomponents.

EXAMPLE 3.11 (PURE AND GEOMETRIC VARIATIONS). Two important exam-ples for such a good variation of mixed Hodge structures are the following:

(i) A polarizable variation ofpureHodge structures is always admissible by adeep theorem of Schmid [46, Theorem 6.16]. So it is good precisely when ithas quasi-unipotent monodromy at infinity.

(ii) Consider a morphismf W X ! Y of complex algebraic varieties withYsmooth, which is a topological fibration with possible singular or noncompactfiber. The locally constant direct image sheavesRnf�QX andRnf!QX (n�0) are good variations of mixed Hodge structures (compare Remark 4.4).

This class of good variations onM is again an abelian categoryVmHsg.M /

stable under tensor product, duality . � /_ and pullbackf � for f an algebraicmorphism of complex algebraic manifolds. Moreover, in thiscase all vectorbundlesFpL of the Hodge filtration carry the structure of a unique under-lying complex algebraic vector bundle (in the Zariski topology), so that thecharacteristic class transformationMHCy can be seen as a natural contravarianttransformation of rings with involution

MHCy WK0.VmHsg.M //!K0alg.M /Œy˙1�:

438 JORG SCHURMANN

In fact, consider a (partial) compactificationM of M as above, withD WDM nM a normal crossing divisor with smooth irreducible components andj W

M ! M the open inclusion. Then the holomorphic vector bundleL withintegrable connectionr corresponding toL has a uniquecanonical Deligneextension.L;r/ to a holomorphic vector bundleL on M , with meromorphicintegrable connection

r W L! L˝OM˝1

M.log.D// (3-10)

having logarithmic polesalongD. Here theresiduesof r alongD have realeigenvalues, sinceL hasquasi-unipotent monodromyalongD. And the canon-ical extension is characterized by the property that all these eigenvalues are inthe half-open intervalŒ0; 1/. Moreover, also the Hodge filtrationF of L extendsuniquely to a filtrationF of L by holomorphic subvector bundles

FpL WD j�.F

pL/\L� j�L;

sinceL is admissiblealongD. Finally, Griffiths transversality extends to

r.FpL/� Fp�1

L˝OM˝1

M.log.D// for all p. (3-11)

For more details see [19, Proposition 5.4] and [38,~ 11.1, 14.4].If we chooseM as a compact algebraic manifold, then we can apply Serre’s

GAGA theorem to conclude thatL and allFpL arealgebraicvector bundles,with r analgebraicmeromorphic connection.

REMARK 3.12. The canonical Deligne extensionL (as above) with its Hodgefiltration F has the following compabilities (compare [19, Part II]):

SMOOTH PULLBACK: Let f WM 0!M be a smooth morphism so thatD0 WD

f �1.D/ is also a normal crossing divisor with smooth irreducible compo-nents onM 0 with complementM 0. Then one has

f �.L/' f �L and f �.FpL/' Fpf �L for all p. (3-12)

EXTERIOR PRODUCT: Let L and L0 be two good variations onM and M 0.Then their canonical Deligne extensions satisfy

LÔM �M 0 L0 ' LÔ

M�M 0L0;

since the residues of the corresponding meromorphic connections are com-patible. Then one has for allp

Fp.LÔM �M 0 L0/'

M

iCkDp

.F iL/Ô

M�M 0.Fk

L0/: (3-13)


TENSOR PRODUCT: In general the canonical Deligne extensions of two goodvariationsL andL0 on M arenot compatible with tensor products, becauseof the choice of different residues for the corresponding meromorphic con-nections. This problem doesn’t appear if one of these variations, lets sayL0,is already defined onM . Let L andL0 be a good variation onM andM ,respectively. Then their canonical Deligne extensions satisfy

L˝OM.L0jM /' L˝O

ML

0;

and one has for allp:

Fp.L˝OM.L0jM //'

M

iCkDp

.F iL/˝O

M.Fk

L0/: (3-14)

Let M be a partial compactification ofM as before, i.e., we don’t assume thatM is compact, withm WD dimC.M /. Then thelogarithmic de Rham complex

DRlog.L/ WD ŒLr

��! � � �r

��! L˝OM˝m

M.log.D//�

(with L in degree zero) is by [19] quasi-isomorphic toRj�L, so that

H �.M;L/'H ��

M ;DRlog.L/�

:

So these cohomology groups get an induced (decreasing)F -filtration comingfrom the filtration

FpDRlog.L/D ŒFpL

r

��! � � �r

��! Fp�mL˝OM˝m

M.log.D//�:

(3-15)For M a compact algebraic manifold, this is again the Hodge filtration of an

induced mixed Hodge structure onH �.M;L/ (compare with Corollary 4.7).

THEOREM 3.13. AssumeM is a smooth algebraic compactification of the al-gebraic manifoldM with the complementD a normal crossing divisor withsmooth irreducible components. Let .L;F;W / be a good variation of mixedHodge structures onM . ThenH n.M;L/'H �

�

M ;DRlog.L/�

gets an inducedmixed Hodge structure withF the Hodge filtration. Moreover, the correspond-ing Hodge to de Rham spectral sequencedegenerates atE1 so that

GrpF.H n.M;L//'H n

�

M;GrpF

DRlog.L/�

for all n;p.

440 JORG SCHURMANN

Therefore one gets as a corollary (compare [12; 13; 35]):

�y.H�.M;L//D

X

n;p

.�1/n �dimC

�

H n�

M;GrpF

DRlog.L/��

� .�y/p

DX

p

��

H ��

M;GrpF

DRlog.L/��

� .�y/p

DX

p;i

.�1/i��

H ��

M;Grp�iF

.L/˝OM˝i

M.log.D//

��

.�y/p

DW hMHCy.Rj�L/; �y

�

˝1

M.log.D//

�

\ ŒOM �i 2 ZŒy˙1�:

(3-16)Here we use the notation

MHCy.Rj�L/ WDX

p

ŒGrpF.L/� � .�y/p 2K0

alg.M /Œy˙1�: (3-17)

Remark 3.12 then implies:

COROLLARY 3.14.LetM be a smooth algebraic partial compactifiction of thealgebraic manifoldM with the complementD a normal crossing divisor withsmooth irreducible components. Then MHCy.Rj�. � // induces a transforma-tion

MHCy.j�. � // WK0.VmHsg.M //!K0alg.M /Œy˙1�:

(1) This is contravariant functorial for a smooth morphismf W M 0 ! M ofsuch partial compactifications, i.e.,

f ��

MHCy.j�. � //�

'MHCy�

j 0

�.f �. � //

�

:

(2) It commutes with exterior products for two good variationsL;L0:

MHCy�

.j � j 0/�Œ.L ˆQM �M 0 L0��

DMHCy.j�ŒL�/ˆ MHCy.j 0

�Œ.L0�/:

(3) Let L be a good variation onM , andL0 one onM . Then MHCy.j�Œ � �/ ismultiplicative in the sense that

MHCy�

j�Œ.L˝QM.L0jM /�

�

DMHCy.j�ŒL�/˝MHCy.ŒL0�/:

4. Calculus of mixed Hodge modules

4A. Mixed Hodge modules. Before discussing extensions of the characteristiccohomology classesMHCy to the singular setting, we need to briefly recallsome aspects of Saito’s theory [39; 40; 41; 43; 44] of algebraic mixed Hodgemodules, which play the role of singular extensions of good variations of mixedHodge structures.


To each complex algebraic varietyZ, Saito associated a categoryMHM.Z/of algebraic mixed Hodge moduleson Z (cf. [39; 40]). If Z is smooth, anobject of this category consists of an algebraic (regular) holonomicD-module.M;F / with a good filtrationF together with a perverse sheafK of rationalvector spaces, both endowed a finite increasing filtrationW such that

˛ WDR.M/an'K˝QZCZ is compatible withW

under the Riemann–Hilbert correspondence coming from the (shifted) analyticde Rham complex (with a chosen isomorphism). Here we use leftD-modules,and the sheafDZ of algebraic differential operators onZ has the increasingfiltration F with FiDZ given by the differential operators of order� i (i 2 Z).Then agoodfiltration F of the algebraic holonomicD-moduleM is given bya bounded from below, increasing and exhaustive filtrationFpM by coherentalgebraicOZ -modules such that

FiDZ .FpM/� FpCiM for all i;p,

and this is an equality fori big enough.(4-1)

In general, for a singular varietyZ one works with suitable local embeddingsinto manifolds and corresponding filteredD-modules supported onZ. In addi-tion, these objects are required to satisfy a long list of complicated properties(not needed here). Theforgetful functor rat is defined as

rat WMHM.Z/! Perv.QZ /; .M.F /;K;W /‘K:

THEOREM 4.1 (M. SAITO). MHM.Z/ is an abelian category with

rat WMHM.Z/! Perv.QZ /

exact and faithful. It extends to a functor

rat WDbMHM.Z/!Dbc .QZ /

to the derived category of complexes ofQ-sheaves with algebraically construc-tible cohomology. There are functors

f�; f!; f�; f !; ˝; ˆ; D onDbMHM.Z/;

which are “lifts” via rat of the similar(derived) functors defined onDbc .QZ /,

with .f �; f�/ and.f!; f!/ also pairs of adjoint functors. One has a natural map

f!! f�, which is an isomorphism forf proper. HereD is a duality involutionD2 ' id “lifting” the Verdier duality functor, with

D ıf � ' f ! ıD and D ıf� ' f! ıD:

442 JORG SCHURMANN

Compare with [40, Theorem 0.1 and~ 4] for more details (as well as with [43]for a more general formal abstraction). The usual truncation �� onDbMHM.Z/corresponds to theperverse truncationp�� on Db

c .Z/ so that

ratıH D pH ı rat;

whereH stands for the cohomological functor inDbMHM.Z/ and pH denotesthe perverse cohomology (always with respect to the self-dual middle perver-sity).

EXAMPLE 4.2. LetM be a complex algebraic manifold of pure complex di-mensionm, with .L;F;W / a good variation of mixed Hodge structures onM .ThenL with its integrable connectionr is a holonomic (left)D-module with˛ WDR.L/an'LŒm�, where this time we use the shifted de Rham complex

DR.L/ WD ŒLr

��! � � �r

��! L˝OM˝m

M�

with L in degree�m, so thatDR.L/an' LŒm� is a perverse sheaf onM . Thefiltration F induces by Griffiths transversality (3-2) a good filtrationFp.L/ WD

F�pL as a filteredD-module. As explained before, this comes from an un-derlying algebraic filteredD-module. Finally is compatible with the inducedfiltration W defined by

W i.LŒm�/ WDW i�mLŒm� and W i.L/ WD .W i�mL/˝QMOM :

And this defines a mixed Hodge moduleM on M , with rat.M/Œ�m� a localsystem onM .

A mixed Hodge moduleM on the purem-dimensional complex algebraic man-ifold M is calledsmoothif rat.M/Œ�m� is a local system onM . Then this ex-ample corresponds to [40, Theorem 0.2], whereas the next theorem correspondsto [40, Theorem 3.27 and remark on p. 313]:

THEOREM 4.3 (M. SAITO). Let M be a purem-dimensional complex alge-braic manifold. Associating to a good variation of mixed Hodge structuresV D .L;F;W / on M the mixed Hodge moduleM WD VH as in Example4.2defines an equivalence of categories

MHM.M /sm ' VmHsg.M /

between the categories of smooth mixed Hodge modules MHM.M /sm and goodvariation of mixed Hodge structures onM . This commutes with exterior productˆ and with the pullbacks

f � WVmHsg.M /!VmHsg.M 0/ and f �Œm0�m� WMHM.M /!MHM.M 0/


for an algebraic morphism of smooth algebraic manifoldsM;M 0 of dimensionm;m0. For M D pt a point, one gets in particular an equivalence

MHM.pt/'mHsp:

REMARK 4.4. These two theorems explain why a geometric variation ofmixedHodge structures as in Example 3.11(2) is good.

By the last identification of the theorem, there exists a unique Tate object

QH .n/ 2MHM.pt/

such that rat.QH .n//DQ.n/ andQH .n/ is of type.�n;�n/:

MHM.pt/ 3QH .n/'Q.n/ 2mHsp:

For a complex varietyZ with constant mapk WZ! pt , define

QHZ .n/ WD k�

QH .n/ 2DbMHM.Z/; with rat.QH

Z .n//DQZ .n/.

So tensoring withQHZ .n/ defines the Tate twist�.n/ of mixed Hodge modules.

To simplify the notation, letQHZ WDQ

HZ .0/. If Z is smoothof complex dimen-

sionn thenQZ Œn� is perverse onZ, andQHZ Œn� 2MHM.Z/ is a single mixed

Hodge module, explicitly described by

QHZ Œn�D ..OZ ;F /;QZ Œn�;W /; with grF

i D 0D grWiCn for all i ¤ 0.

It follows from the definition that everyM2MHM.Z/ has a finite increasingweight filtration W so that the functorM ! GrW

kM is exact. We say that

M2DbMHM.Z/ hasweights�n .resp.�n/ if GrWj H iM D0 for all j >nCi

(resp.j < nC i). M is calledpure of weightn if it has weights both� n and� n. For the following results compare with [40, Proposition 2.26 and (4.5.2)]:

PROPOSITION4.5. If f is a map of algebraic varieties, thenf! andf � preserveweight� n, andf� andf ! preserve weight� n. If f is smooth of pure complexfiber dimensionm, thenf ! ' f �Œ2m�.m/ so thatf �; f ! preserve pure objectsfor f smooth. Moreover, if M2DbMHM.X / is pure andf WX!Y is proper,thenf�M 2DbMHM.Y / is pure of the same weight asM.

Similarly the duality functorD exchanges “weight� n” and “weight ��n” ,in particular it preserves pure objects. Finally let j W U ! Z be the inclusionof a Zariski open subset. Then theintermediate extensionfunctor

j!� WMHM.U /!MHM.Z/ W M‘ Im�

H 0.j!M/!H 0.j�.M/�

(4-2)

preserves weight� n and� n, and so preserves pure objects(of weightn).

444 JORG SCHURMANN

We say thatM 2DbMHM.Z/ is supported onS �Z if and only if rat.M/ issupported onS . There are the abelian subcategoriesMH.Z; k/p � MHM.Z/of pure Hodge modules of weightk, which in the algebraic context are assumedto be polarizable (and extendable at infinity).

For eachk 2 Z, the abelian categoryMH.Z; k/p is semisimple, in the sensethat every pure Hodge module onZ can be uniquely written as a finite direct sumof pure Hodge modules with strict support in irreducible closed subvarieties ofZ. Let MHS .Z; k/

p denote the subcategory ofpure Hodge modules of weightk with strict support inS . Then everyM 2MHS .Z; k/

p is generically a goodvariation of Hodge structuresVU of weight k � d (whered D dimS) on aZariski dense smooth open subsetU � S ; i.e., VU is polarizable with quasi-unipotent monodromy at infinity. This follows from Theorem 4.3 and the factthat a perverse sheaf is generically a shifted local system on a smooth denseZariski open subsetU � S . Conversely, every such good variation of HodgestructuresV on such anU corresponds by Theorem 4.3 to a pure Hodge moduleVH on U , which can be extended in an unique way to a pure Hodge modulej!�VH on S with strict support (herej W U ! S is the inclusion). Under thiscorrespondence, forM 2MHS .Z; k/

p we have that

rat.M/D ICS .V/

is the twisted intersection cohomology complexfor V the corresponding varia-tion of Hodge structures. Similarly

D.j!�VH /' j!�.V_

H /.d/: (4-3)

Moreover, apolarizationof M 2 MHS .Z; k/p corresponds to an isomor-

phism of Hodge modules (compare [38, Definition 14.35, Remark 14.36])

S WM'D.M/.�k/; (4-4)

whose restriction toU gives a polarization ofV. In particular it induces a self-duality isomorphism

S W rat.M/'D.rat.M//.�k/'D.rat.M//

of the underlying twisted intersection cohomology complex, if an isomorphismQU .�k/'QU is chosen.

So if U is smooth of pure complex dimensionn, thenQHU Œn� is a pure Hodge

module of weightn. If moreoverj W U ŒZ is a Zariski-open dense subset inZ, then theintermediate extensionj!� for mixed Hodge modules (cf. also with[7]) preserves the weights. This shows that ifZ is a complex algebraic varietyof pure dimensionn andj W U ŒZ is the inclusion of a smooth Zariski-opendense subset then the intersection cohomology moduleIC H

ZWD j!�.Q

HU Œn�/ is

pure of weightn, with underlying perverse sheaf rat.IC HZ/D ICZ .


Note that the stability of a pure objectM 2 MHM.X / under a proper mor-phism f W X ! Y implies the famousdecomposition theoremof [7] in thecontext of pure Hodge modules [40, (4.5.4) on p. 324]:

f�M'M

i

H if�MŒ�i �; with H if�M semisimple for alli . (4-5)

AssumeY is pure-dimensional, withf WX!Y a resolution of singularities,i.e., X is smooth withf a proper morphism, which generically is an isomor-phism on some Zariski dense open subsetU . ThenQ

HX is pure, sinceX is

smooth, andIC HY

has to be the direct summand ofH 0f�QHX which corresponds

to QHU .

COROLLARY 4.6. AssumeY is pure-dimensional, with f WX ! Y a resolutionof singularities.ThenIC H

Yis a direct summand off�Q

HX 2DbMHM.Y /.

Finally we get the following results about the existence of amixed Hodgestructure on the cohomology (with compact support)H i

.c/.Z;M/ for M 2

DbMHM.Z/.

COROLLARY 4.7. Let Z be a complex algebraic variety with constant mapk W Z ! pt . Then the cohomology(with compact support) H i

.c/.Z;M/ of

M 2DbMHM.Z/ gets an induced graded polarizable mixed Hodge structure:

H i.c/.Z;M/DH i.k�.!/M/ 2MHM.pt/'mHsp:

In particular:

(1) The rational cohomology(with compact support) H i.c/.Z;Q/ of Z gets an

induced graded polarizable mixed Hodge structure by

H i.Z;Q/D rat.H i.k�k�Q

H // and H ic .Z;Q/D rat.H i.k!k

�Q

H //:

(2) Let VU be a good variation of mixed Hodge structures on a smooth puren-dimensional complex varietyU , which is Zariski open and dense in a varietyZ, with j W U ! Z the open inclusion. Then the global twisted intersectioncohomology(with compact support)

IH i.c/.Z;V/ WDH i

.c/

�

Z; ICZ .V/Œ�n��

gets a mixed Hodge structure by

IH i.c/.Z;V/DH i

�

k�.!/ICZ .V/Œ�n��

DH i�

k�.!/j!�.V/Œ�n��

:

If Z is compact, with V a polarizable variation of pure Hodge structures ofweightw, then alsoIH i.Z;V/ has a(polarizable) pure Hodge structure ofweightwC i .

446 JORG SCHURMANN

(3) Let V be a good variation of mixed Hodge structures on a smooth(pure-dimensional) complex manifoldM , which is Zariski open and dense in com-plex algebraic manifoldM , with complementD a normal crossing divisorwith smooth irreducible components. ThenH i.M;V/ gets a mixed Hodgestructure by

H i.M;V/'H i.M ; j�V/'H i.k�j�V/;

with j W U !Z the open inclusion.

REMARK 4.8. Here are important properties of these mixed Hodge structures:

(1) By a deep theorem of Saito [44, Theorem 0.2, Corollary 4.3], the mixedHodge structure onH i

.c/.Z;Q/ defined as above coincides with the classical

mixed Hodge structure constructed by Deligne ([20; 21]).(2) Assume we are in the context of Corollary 4.7(3) withZDM projective and

V a good variation of pure Hodge structures onU DM . Then the pure Hodgestructure of (2) on the global intersection cohomologyIH i.Z;V/ agrees withthat of [15; 29] defined in terms ofL2-cohomology with respect to a Kahlermetric with Poincare singularities alongD (compare [40, Remark 3.15]). Thecase of a1-dimensional complex algebraic curveZ DM due to Zucker [56,Theorem 7.12] is used in the work of Saito [39, (5.3.8.2)] in the proof of thestability of pure Hodge modules under projective morphisms[39, Theorem5.3.1] (compare also with the detailed discussion of this1-dimensional casein [45]).

(3) Assume we are in the context of Corollary 4.7(3) withM compact. Thenthe mixed Hodge structure onH i.M;V/ is the one of Theorem 3.13, whoseHodge filtrationF comes from the filtered logarithmic de Rham complex(compare [40,~ 3.10, Proposition 3.11]).

4B. Grothendieck groups of algebraic mixed Hodge modules. In this section,we describe the functorial calculus of Grothendieck groupsof algebraic mixedHodge modules. LetZ be a complex algebraic variety. By associating to (theclass of) a complex the alternating sum of (the classes of) its cohomology ob-jects, we obtain the following identification (compare, forexample, [30, p. 77]and [47, Lemma 3.3.1])

K0.DbMHM.Z//DK0.MHM.Z//: (4-6)

In particular, ifZ is a point, then

K0.DbMHM.pt//DK0.mHsp/; (4-7)


and the latter is a commutative ring with respect to the tensor product, with unitŒQH �. Then we have, for any complexM� 2DbMHM.Z/, the identification

ŒM��DX

i2Z

.�1/i ŒH i.M�/� 2K0.DbMHM.Z//ŠK0.MHM.Z//: (4-8)

In particular, if for anyM2MHM.Z/ andk 2Z we regardMŒ�k� as a complexconcentrated in degreek, then

ŒMŒ�k��D .�1/k ŒM� 2K0.MHM.Z//: (4-9)

All the functorsf�, f!, f �, f !, ˝, ˆ, D induce corresponding functors onK0.MHM. � //. Moreover,K0.MHM.Z// becomes aK0.MHM.pt//-module,with the multiplication induced by the exact exterior product with a point space:

ˆ WMHM.Z/�MHM.pt/!MHM.Z � fptg/'MHM.Z/:

Also note that

M˝QHZ 'Mˆ Q

Hpt 'M

for all M 2MHM.Z/. Therefore,K0.MHM.Z// is a unitaryK0.MHM.pt//-module. The functorsf�, f!, f �, f ! commute with exterior products (andf �

also commutes with the tensor product˝), so that the induced maps at the levelof Grothendieck groupsK0.MHM. � // areK0.MHM.pt//-linear. SimilarlyDdefines an involution onK0.MHM. � //. Moreover, by the functor

rat WK0.MHM.Z//!K0.Dbc .QZ //'K0.Perv.QZ //;

all these transformations lift the corresponding transformations from the (topo-logical) level of Grothendieck groups of constructible (orperverse) sheaves.

REMARK 4.9. The Grothendieck groupK0.MHM.Z// has two different typesof generators:

(1) It is generated by the classes of pure Hodge modulesŒICS .V/� with strictsupport in an irreducible complex algebraic subsetS � Z, with V a goodvariation of (pure) Hodge structures on a dense Zariski opensmooth subsetU of S . These generators behave well under duality.

(2) It is generated by the classesf�Œj�V�, with f WM !Z a proper morphismfrom the smooth complex algebraic manifoldM , j WM !M the inclusionof a Zariski open and dense subsetM , with complementD a normal crossingdivisor with smooth irreducible components, andV a good variation of mixed(or if one wants also pure) Hodge structures onM . These generators will beused in the next section about characteristic classes of mixed Hodge modules.

448 JORG SCHURMANN

Here (1) follows from the fact that a mixed Hodge module has a finite weightfiltration, whose graded pieces are pure Hodge modules, i.e., are finite directsums of pure Hodge modulesICS .V/ with strict supportS as above. Theclaim in (2) follows by induction from resolution of singularities and from theexistence of a “standard” distinguished triangle associated to a closed inclusion.

Let i W Y !Z be a closed inclusion of complex algebraic varieties with opencomplementj W U D ZnY ! Z. Then one has by Saito’s work [40, (4.4.1)]the following functorial distinguished triangle inDbMHM.Z/:

j!j�

adj

��! idadi

��! i�i�Œ1��! : (4-10)

Here the mapsad are the adjunction maps, withi� D i! sincei is proper. Iff W Z ! X is a complex algebraic morphism, then we can applyf! to getanother distinguished triangle

f!j!j�Q

HZ

adj

��! f!QHZ

adi

��! f!i!i�Q

HZ

Œ1��! : (4-11)

On the level of Grothendieck groups, we get the importantadditivity relation

f!ŒQHZ �D .f ı j /!ŒQ

HU �C .f ı i/!ŒQ

HY �

2K0.DbMHM.X //DK0.MHM.X //: (4-12)

COROLLARY 4.10.One has a natural group homomorphism

�Hdg WK0.var=X /!K0.MHM.X //I Œf WZ!X �‘ Œf!QHZ �;

which commutes with pushdownf!, exterior product and pullbackg�. ForX D pt this corresponds to the ring homomorphism(2-10)under the identifi-cation MHM.pt/'mHsp.

HereK0.var=X / is the motivicrelative Grothendieck groupof complex alge-braic varieties overX , i.e., the free abelian group generated by isomorphismclassesŒf �D Œf WZ! X � of morphismsf to X , divided out be theadditivityrelation

Œf �D Œf ı i �C Œf ı j �

for a closed inclusioni WY !Z with open complementj WU DZnY !Z. Thepushdownf!, exterior product and pullbackg� for these relative Grothendieckgroups are defined by composition, exterior product and pullback of arrows. Thefact that�Hdg commutes with exterior product (or pullbackg�) follows thenfrom the corresponding Kunneth (or base change) theorem for the functor

f! WDbMHM.Z/!DbMHM.X /

(contained in Saito’s work [43] and [40, (4.4.3)]).


Let L WD ŒA1C � 2K0.var=pt/ be the class of the affine line, so that

�Hdg.L/D ŒH2.P 1.C/;Q/�D ŒQ.�1/� 2K0.MHM.pt//DK0.mHsp/

is the Lefschetz classŒQ.�1/�. This class is invertible inK0.MHM.pt// D

K0.mHsp/ so that the transformation�Hdg of Corollary 4.10 factorizes over thelocalization

M0.var=X / WDK0.var=X /ŒL�1�:

Altogether we get the following diagram of natural transformations commutingwith f!, ˆ andg�:

F.X /can �� M0.var=X / �� K0.var=X /

�stalk

x

?

?

?

?

y

�Hdg

K0.Dbc .X // ��rat

K0.MHM.X //:

(4-13)

HereF.X / is the group of algebraically constructible functions onX , whichis generated by the collectionf1Z g, for Z �X a closed complex algebraic sub-set, with�stalk given by the Euler characteristic of the stalk complexes (compare[47, ~ 2.3]). The pushdownf! for algebraically constructible functions is definedfor a morphismf W Y ! X by

f!.1Z /.x/ WD ��

H �

c .Z \ff D xg;Q/�

for x 2X ,

so that the horizontal arrow marked “can” is given by

canW Œf W Y ! X �‘ f!.1Y /; with can.L/D 1pt .

The advantage ofM0.var=X / compared toK0.var=X / is that it has an in-ducedduality involutionD WM0.var=X /!M0.var=X / characterized uniquelyby the equality

D .Œf WM ! X �/D L�m � Œf WM !X �

for f WM ! X a proper morphism withM smooth and purem-dimensional(compare [8]). This “motivic duality”D commutes with pushdownf! for properf , so that�Hdg also commutes with duality by

�Hdg .DŒidM �/D �Hdg�

L�m � ŒidM �

�

D ŒQHM .m/�

D ŒQHM Œ2m�.m/�D ŒD.QH

M /�DD�

�Hdg .ŒidM �/�

(4-14)

for M smooth and purem-dimensional. In fact by resolution of singularities and“additivity”, K0.var=X / is generated by such classesf!ŒidM �D Œf WM !X �.

Then all the transformations in the diagram (4-13)commute with duality,were K0.D

bc .X // gets this involution from Verdier duality, andD D id for

algebraically constructible functions by can.ŒQ.�1/�/D1pt (compare also with

450 JORG SCHURMANN

[47, ~ 6.0.6]). Similarly they commute withf� andg! defined by the relations(compare [8]):

D ıg� D g! ıD and D ıf� D f! ıD:

For example for an open inclusionj WM !M , one gets

�Hdg .j�ŒidM �/D j�ŒQHM �: (4-15)

5. Characteristic classes of mixed Hodge modules

5A. Homological characteristic classes. In this section we explain the theoryof K-theoretical characteristic homology classes of mixed Hodge modules basedon the following result of Saito (compare with [39,~ 2.3] and [44,~ 1] for thefirst part, and with [40,~ 3.10, Proposition 3.11]) for part (2)):

THEOREM 5.1 (M. SAITO). Let Z be a complex algebraic variety. Then thereis a functor of triangulated categories

GrFp DR WDbMHM.Z/!Db

coh.Z/ (5-1)

commuting with proper push-down, with GrFp DR.M/ D 0 for almost all p

and M fixed, whereDbcoh.Z/ is the bounded derived category of sheaves of

algebraicOZ -modules with coherent cohomology sheaves. If M is a (purem-dimensional) complex algebraic manifold, then one has in addition:

(1) LetM 2 MHM.M / be a single mixed Hodge module. ThenGrFp DR.M/

is the corresponding complex associated to the de Rham complex of the un-derlying algebraic leftD-moduleM with its integrable connectionr:

DR.M/D ŒMr

��! � � �r

��! M˝OM˝m

M�

with M in degree�m, filtered by

FpDR.M/D ŒFpMr

��! � � �r

��! FpCmM˝OM˝m

M�:

(2) Let M be a smooth partial compactification of the complex algebraic man-ifold M with complementD a normal crossing divisor with smooth irre-ducible components, with j WM!M the open inclusion. LetVD .L;F;W /

be a good variation of mixed Hodge structures onM . Then the filtered deRham complex

.DR.j�V/;F / of j�V 2MHM.M /Œ�m��DbMHM.M /

is filtered quasi-isomorphic to the logarithmic de Rham complex DRlog.L/

with the increasing filtrationF�p WDFp (p2Z) associated to the decreasing


F -filtration (3-15). In particular GrF�pDR.j�V/ (p2Z) is quasi-isomorphic

to

GrpF

DRlog.L/D ŒGrpFL

Gr r

��! � � �Gr r

��! Grp�mF

L˝OM˝m

M.log.D//�:

Here the filtrationFpDR.M/ of the de Rham complex is well defined, sincethe action of the integrable connectionr is given in local coordinates.z1; : : : ;

zm/ by

r. � /D

mX

iD1

@

@zi. � /˝ dzi ; with

@

@zi2 F1DM ;

so thatr.FpM/ � FpC1M for all p by (4-1). For later use, let us point thatthe mapsGr r andGr r in the complexes

GrFp DR.M/ and Grp

FDRlog.L/

areO-linear!

EXAMPLE 5.2. LetM be a purem-dimensional complex algebraic manifold.Then

GrF�pDR.QH

M /'˝pMŒ�p� 2Db

coh.M /

if 0 � p � m, andGrF�pDR.QH

M / ' 0 otherwise. Assume in addition thatf W M ! Y is a resolution of singularities of the pure-dimensional complexalgebraic varietyY . ThenIC H

Yis a direct summand off�Q

HM 2DbMHM.Y /

so that by functorialitygrF�pDR.IC H

Y/ is a direct summand ofRf�˝

pMŒ�p�2

Dbcoh.Y /. In particular

GrF�pDR.IC H

Y /' 0 for p < 0 or p >m.

The transformationsGrFp DR (p 2 Z) induce functors on the level of Grothen-

dieck groups. Therefore, ifG0.Z/ ' K0.Dbcoh.Z// denotes the Grothendieck

group of coherentalgebraicOZ -sheaves onZ, we get group homomorphisms

GrFp DR WK0.MHM.Z//DK0.D

bMHM.Z//!K0.Dbcoh.Z//'G0.Z/:

DEFINITION 5.3. Themotivic Hodge Chern class transformation

MHCy WK0.MHM.Z//! G0.Z/˝ZŒy˙1�

is defined by

ŒM�‘X

i;p

.�1/i ŒHi.GrF�pDR.M//� � .�y/p: (5-2)

452 JORG SCHURMANN

So this characteristic class captures information from thegraded pieces of thefiltered de Rham complex of the filteredD-module underlying a mixed HodgemoduleM 2MHM.Z/, instead of the graded pieces of the filteredD-moduleitself (as more often studied). Letp0 D minfp j FpM ¤ 0g. Using Theorem5.1(1) for a local embeddingZŒM of Z into a complex algebraic manifoldM of dimensionm, one gets

GrFp DR.M/D 0 for p < p0�m;

and

GrFp0�mDR.M/' .Fp0M/˝OM

!M

is a coherentOZ -sheaf independent of the local embedding. Here we are us-ing left D-modules (related to variation of Hodge structures), whereas for thisquestion the corresponding filtered rightD-module (as used in [42])

Mr WDM˝OM

!M with FpMr WD

�

FpCmM�

˝OM!M

would work better. Then the coefficient of the “top-dimensional” power ofy in

MHCy.ŒM�/D ŒFp0M˝OM!M �˝.�y/m�p0

CX

i<m�p0

. � � � / �yi 2G0.Z/Œy˙1�

(5-3)

is given by the classŒFp0M ˝OM!M � 2 G0.Z/ of this coherentOZ -sheaf

(up to a sign). Using resolution of singularities, one gets for example for anm-dimensional complex algebraic varietyZ that

MHCy.ŒQHZ �/D Œ��!M � �ymC

X

i<m

. � � � / �yi 2G0.Z/Œy˙1�;

with � WM ! Z any resolution of singularities ofZ (compare [44, Corollary0.3]). More generally, for an irreducible complex varietyZ andMD IC H

Z.L/

a pure Hodge module with strict supportZ, the corresponding coherentOZ -sheaf

SZ .L/ WD Fp0IC HZ .L/˝OM

!M

only depends onZ and the good variation of Hodge structuresL on a Zariskiopen smooth subset ofZ, and it behaves much like a dualizing sheaf. Its formalproperties are studied in Saito’s proof given in [42] of a conjecture of Kollar. Sothe “top-dimensional” power ofy in MHCy

�

ŒIC HZ.L/�

�

exactly picks out (upto a sign) the classŒSZ .L/� 2 G0.Z/ of this interesting coherent sheafSZ .L/

on Z.Let td.1Cy/ be thetwisted Todd transformation

td.1Cy/ WG0.Z/˝ZŒy˙1�!H�.Z/˝QŒy˙1; .1Cy/�1� I

ŒF �‘X

k�0

tdk.ŒF �/ � .1Cy/�k ; (5-4)


whereH�. � / stands either for the Chow homology groupsCH�. � / or for theBorel–Moore homology groupsH BM

2�. � / (in even degrees), andtdk is the de-

greek component inHk.Z/ of theTodd class transformationtd� W G0.Z/!

H�.Z/˝Q of Baum, Fulton, and MacPherson [5], which is linearly extendedoverZŒy˙1�. Compare also with [22, Chapter 18] and [24, Part II].

DEFINITION 5.4. The (un)normalizedmotivic Hirzebruch class transforma-tions MHTy� (andAMHTy�) are defined by the composition

MHTy� WD td.1Cy/ ıMHCy WK0.MHM.Z//!H�.Z/˝QŒy˙1; .1Cy/�1�

(5-5)and

AMHTy� WD td� ıMHCy WK0.MHM.Z//!H�.Z/˝QŒy˙1�: (5-6)

REMARK 5.5. By precomposing with the transformation�Hdg from Corollary4.10 one gets similar transformations

mCy WDMHCy ı�Hdg; Ty� WDMHTy� ı�Hdg; QTy� WDAMHTy� ı�Hdg

defined on the relative Grothendieck group of complex algebraic varieties asstudied in [9]. Then it is the (normalized) motivic Hirzebruch class transfor-mationTy�, which, as mentioned in the Introduction, “unifies” in a functorialway

.yD�1/ the (rationalized) Chern class transformationc� of MacPherson [34];

.y D 0/ the Todd class transformationtd� of Baum–Fulton–MacPherson [5];

.y D 1/ theL-class transformationL� of Cappell and Shaneson [14].

(Compare with [9; 48] and also with [55] in these proceedings.)

In this paper we work most the time only with the more important K-theoreticaltransformationMHCy . The corresponding results forMHTy� follow from thisby the known properties of the Todd class transformationtd� (compare [5; 22;24]).

EXAMPLE 5.6. LetVD .V;F;W / 2MHM.pt/DmHsp be a (graded polariz-able) mixed Hodge structure. Then:

MHCy.ŒV�/DX

p

dimC.GrpF

VC/ � .�y/p D �y.ŒV�/ 2 ZŒy˙1�

DG0.pt/˝ZŒy˙1�: (5-7)

So over a point the transformationMHCy coincides with the�y-genus ringhomomorphism�y W K0.mHsp/ ! ZŒy˙1� (and similarly for AMHTy� andMHTy�).

454 JORG SCHURMANN

Themotivic Chern classCy.Z/ and themotivic Hirzebruch classTy�.Z/ of acomplex algebraic varietyZ are defined by

Cy.Z/ WDMHCy.ŒQHZ �/ and Ty�.Z/ WDMHTy�.ŒQ

HZ �/: (5-8)

Similarly, if U is a puren-dimensional complex algebraic manifold andL is alocal system onU underlying a good variation of mixed Hodge structuresL,we define thetwisted motivic Chern and Hirzebruch characteristic classesby(compare [12; 13; 35])

Cy.U IL/ WDMHCy.ŒLH �/ and Ty�.U IL/ WDMHTy�.ŒL

H �/; (5-9)

whereLH Œn� is the smooth mixed Hodge module onU with underlying perversesheafLŒn�. Assume, in addition, thatU is dense and Zariski open in the complexalgebraic varietyZ. Let IC H

Z; IC H

Z.L/ 2MHM.Z/ be the (twisted) intersec-

tion homology (mixed) Hodge module onZ, whose underlying perverse sheaf isICZ or ICZ .L/, as the case may be. Then we defineintersection characteristicclassesas follows (compare [9; 11; 13; 35]):

ICy.Z/ WDMHCy

��

IC HZ Œ�n�

��

;

ITy�.Z/ WDMHTy�

��

IC HZ Œ�n�

��

;(5-10)

and, similarly,

ICy.ZIL/ WDMHCy

��

IC HZ .L/Œ�n�

��

;

ITy�.ZIL/ WDMHTy�

��

IC HZ .L/Œ�n�

��

:(5-11)

By definition and Theorem 5.1, the transformationsMHCy andMHTy� com-mute with proper push-forward. The following normalizationproperty holds(compare [9]): IfM is smooth, then

Cy.Z/D �y.T�M /\ ŒOM � and Ty�.Z/D T �

y .TM /\ ŒM �; (5-12)

whereT �y .TM / is the cohomology Hirzebruch class ofM as in Theorem 2.4.

EXAMPLE 5.7. LetZ be a compact (possibly singular) complex algebraic va-riety, with k W Z ! pt the proper constant map to a point. Then forM 2

DbMHM.Z/ the pushdown

k�.MHCy.M//DMHCy.k�M/D �y

�

ŒH �.Z;M/��

is the Hodge genus

�y.ŒH�.Z;M/�/D

X

i;p

.�1/i dimC.GrpF

H i.Z;M// � .�y/p: (5-13)

In particular:


(1) If Z is smooth, then

k�Cy.Z/D �y.Z/ WD �y

�

ŒH �.Z;Q/��

k�Cy.ZIL/D �y.ZIL/ WD �y

�

ŒH �.Z;L/��

:

(2) If Z is pure-dimensional, then

k�ICy.Z/D I�y.Z/ WD �y

�

ŒIH �.Z;Q/��

k�ICy.ZIL/D I�y.ZIL/ WD �y

�

ŒIH �.Z;L/��

:

Note that, forZ compact,

I��1.Z/D �.ŒIH �.ZIQ/�

is theintersection(co)homology Euler characteristicof Z, whereas, forZ pro-jective,

I�1.Z/D sgn�

IH �.Z;Q/�

is the intersection(co)homology signatureof Z, introduced by Goresky andMacPherson [25]. In fact this follows as in the smooth context from Saito’srelative version of the Hodge index theorem for intersection cohomology [39,Theorem 5.3.2]. Finally�0.Z/ and I�0.Z/ are two possible extensions tosingular varieties of thearithmetic genus. Here it makes sense to takey D 0,since one has, by Example 5.2,

k�ICy.Z/D I�y.Z/ 2 ZŒy�:

It is conjectured that, for a puren-dimensional compact varietyZ,

IT1�.Z/?DL�.Z/ 2H2�.Z;Q/

is the Goresky–MacPherson homologyL-class [25] of the Witt spaceZ; see[9, Remark 5.4]. Similarly one should expect for a pure-dimensional compactvarietyZ that

˛.IC1.Z//?D4.Z/ 2KO

top0.Z/

�

12

�

˚KOtop2.Z/

�

12

�

'Ktop0.Z/

�

12

�

; (5-14)

where WG0.Z/!Ktop0.Z/ is theK-theoretical Riemann–Roch transformation

of Baum, Fulton, and MacPherson [6], and4.Z/ is theSullivan classof the WittspaceZ (compare with [3] in these proceedings). These conjecturedequalitiesare true for a smoothZ, or more generally for a puren-dimensional compactcomplex algebraic varietyZ with asmall resolutionof singularitiesf WM!Z,in which case one hasf�.Q

HM /D IC H

ZŒ�n�, so that

IT1�.Z/D f�T1�.M /D f�L�.M /DL�.Z/

and

˛ .IC1.Z//D f� .˛.C1.M ///D f�4.M /D4.Z/:

456 JORG SCHURMANN

Here the functorialityf�L�.M /D L�.Z/ andf�4.M /D4.Z/ for a smallresolution follows, for instance, from [54], which allows one to think of thecharacteristic classesL� and4 as covariant functors for suitable Witt groupsof selfdual constructible sheaf complexes.

In particular, the classesf�C1.M / and f�T1�.M / do not depend on thechoice of a small resolution. In fact the same functorialityargument applies to

ICy.Z/D f�Cy.M / 2G0.Z/˝ZŒy�;

ITy�.Z/D f�Ty�.M / 2H2�.Z/˝QŒy; .1Cy/�1�I

compare [11; 35]. Note that in general a complex varietyZ doesn’t have asmall resolution, and even if it exists, it is in general not unique. This type ofindependence question were discussed by Totaro [51], pointing out the relationto the famouselliptic genus and classes(compare also with [32; 53] in theseproceedings). Note that we get such a result for theK-theoretical class

ICy.Z/D f�Cy.M / 2G0.Z/˝ZŒy� !

5B. Calculus of characteristic classes. So far we only discussed the functorial-ity of MHCy with respect to proper push down, and the corresponding relationto Hodge genera for compactZ coming from the push down for the properconstant mapk WZ! pt . Now we explain some other important functorialityproperties. Their proof is based on the following (see [35, (4.6)], for instance):

EXAMPLE 5.8. LetM be a smooth partial compactification of the complex alge-braic manifoldM with complementD a normal crossing divisor with smooth ir-reducible components, withj WM !M the open inclusion. LetVD .L;F;W /

be a good variation of mixed Hodge structures onM . Then the filtered de Rhamcomplex

.DR.j�V/;F / of j�V 2MHM.M /Œ�m��DbMHM.M /

is by Theorem 5.1(2) filtered quasi-isomorphic to the logarithmic de Rham com-plex DRlog.L/ with the increasing filtrationF�p WD Fp (p 2 Z) associated tothe decreasingF -filtration (3-15). Then

MHCy.j�V/DX

i;p

.�1/i ŒHi.GrpF

DRlog.L//� � .�y/p

DX

p

ŒGrpF

DRlog.L/� � .�y/p

.�/DX

i;p

.�1/i ŒGrp�iF

.L/˝OM˝i

M.log.D//� � .�y/p

DMHCy.Rj�L/\�

�y

�

˝1M.log.D//

�

\ ŒOM ��

: (5-15)


In particular forj D id W M ! M we get the followingAtiyah–Meyer typeformula (compare [12; 13; 35]):

MHCy.V/DMHCy.L/\�

�y.T�M /\ ŒOM �

�

: (5-16)

REMARK 5.9. The formula (5-15) is a class version of the formula (3-16) ofTheorem 3.13, which one gets back from (5-15) by pushing downto a point forthe proper constant mapk WM ! pt on the compactificationM of M .

Also note that in the equality (�) in (5-15) we use the fact that the com-plex Grp

FDRlog.L/ has coherent (locally free) objects, withOM -linear maps

between them.

The formula (5-15) describes asplitting of the characteristic classMHCy.j�V/

into two terms:

(coh) a cohomological termMHCy.Rj�L/, capturing the information of thegood variation of mixed Hodge structuresL, and

(hom) the homological term�y

�

˝1M.log.D//

�

\ ŒOM �DMHCy.j�QHM /, cap-

turing the information of the underlying space or embeddingj WM !M .

By Corollary 3.14, the termMHCy.Rj�L/ has good functorial behavior withrespect to exterior and suitable tensor products, as well asfor smooth pullbacks.For the exterior products one gets similarly (compare [19, Proposition 3.2]):

˝1M �M 0.log.D �M 0[M �D0//'

�

˝1M.log.D//

�

ˆ�

˝1M 0.log.D0//

�

so that

�y

�

˝1M�M 0.log.D �M 0[M �D0//

�

\ ŒOM�M 0 �

D�

�y

�

˝1M.log.D//

�

\ ŒOM ��

ˆ�

�y

�

˝1M 0.log.D0//

�

\ ŒOM 0 ��

for the product of two partial compactifications as in example 5.8. But theGrothendieck groupK0.MHM.Z// of mixed Hodge modules on the complexvarietyZ is generated by classes of the formf�.j�ŒV�/, with f WM!Z properandM;M ;V as before. Finally one also has the multiplicativity

.f � f 0/� D f� ˆ f 0

�

for the push down for proper mapsf WM !Z andf 0 WM 0!Z0 on the level ofGrothendieck groupsK0.MHM. � // as well as forG0. � /˝ZŒy˙1�. Then onegets the following result from Corollary 3.14 and Example 5.8 (as in [9, Proofof Corollary 2.1(3)]):

458 JORG SCHURMANN

COROLLARY 5.10 (MULTIPLICATIVITY FOR EXTERIOR PRODUCTS). Themotivic Chern class transformation MHCy commutes with exterior products:

MHCy.ŒM ˆ M 0�/DMHCy.ŒM �ˆ ŒM 0�/

DMHCy.ŒM �/ˆ MHCy.ŒM0�/ (5-17)

for M 2DbMHM.Z/ andM 0 2DbMHM.Z0/.

Next we explain the behavior ofMHCy for smooth pullbacks. Consider a carte-sian diagram of morphisms of complex algebraic varieties

M 0g0

��! M

f 0

?

?

y

?

?

yf

Z0 ��!g

Z;

with g smooth,f proper andM;M ;V as before. Theng0 too is smooth andf 0 is proper, and one has thebase change isomorphism

g�f� D f0

�g0�

on the level of Grothendieck groupsK0.MHM. � // as well as forG0. � / ˝

ZŒy˙1�. Finally for the induced partial compactificationM 0 of M 0 WDg0�1.M /,with complementD0 the induced normal crossing divisor with smooth irre-ducible components, one has a short exact sequence of vectorbundles onM 0:

0! g0��

˝1M.log.D//

�

!˝1M 0.log.D0//! T �

g0 ! 0;

with T �

g0 the relative cotangent bundle along the fibers of the smooth morphismg0. And by base change one hasT �

g0 D f 0�.T �g /. So for the corresponding

lambda classes we get

�y

�

˝1M 0.log.D0//

�

D�

g0��y

�

˝1M.log.D//

��

˝�y.T�

g0/

D�

g0��y

�

˝1M.log.D//

��

˝ f 0��y.T�

g /:(5-18)

Finally (compare also with [9, Proof of Corollary 2.1(4)]),by using thepro-jection formula

�y.T�

g /˝ f0

�. � /D f 0

�

�

f 0��y.T�

g /˝ . � /�

W

G0.M0/˝ZŒy˙1�!G0.Z

0/˝ZŒy˙1�

one gets from Corollary 3.14 and Example 5.8 the following consequence:


COROLLARY 5.11 (VRRFOR SMOOTH PULLBACKS). For a smooth morphismg WZ0!Z of complex algebraic varieties one has for the motivic Chernclasstransformation the following Verdier Riemann–Roch formula:

�y.T�

g /\g�MHCy.ŒM�/DMHCy.g�ŒM�/DMHCy.Œg

�M�/ (5-19)

for M 2DbMHM.Z/. In particular

g�MHCy.ŒM�/DMHCy.g�ŒM�/DMHCy.Œg

�M�/ (5-20)

for g anetale morphism(i.e., a smooth morphism with zero dimensional fibers),or in more topological terms, for g an unramified covering. The most importantspecial case is that of an open embedding.

If moreoverg is also proper, then one gets from Corollary 5.11 and the projectionformula the following result:

COROLLARY 5.12 (GOING UP AND DOWN). Let g WZ0!Z be a smooth andproper morphism of complex algebraic varieties. Then one has for the motivicChern class transformation the following going up und down formula:

MHCy.g�g�ŒM�/D g�MHCy.g�ŒM�/

D g�

�

�y.T�

g /\g�MHCy.ŒM�/�

D�

g��y.T�

g /�

\MHCy.ŒM�/ (5-21)

for M 2DbMHM.Z/, with

g�

�

�y.T�

g /�

WDX

p;q�0

.�1/q � ŒRqg�.˝p

Z 0=Z/� �yp 2K0

alg.Z/Œy�

the algebraic cohomology class being given(as in Example3.5)by

MHCy.ŒRg�QZ 0 �/DX

p;q�0

.�1/q � ŒRqg�.˝p

Z 0=Z/� �yp:

Note that all higher direct image sheavesRqg�.˝p

Z 0=Z/ are locally free in this

case, sinceg is a smooth and proper morphism of complex algebraic varieties(compare with [18]).In particular

g�Cy.Z0/D

�

g��y.T�

g /�

\Cy.Z/;

and

g�ICy.Z0/D

�

g��y.T�

g /�

\ ICy.Z/

for Z and Z0 pure-dimensional. If , in addition, Z and Z0 are compact, withk WZ! pt the constant proper map, then

�y.g�ŒM�/D k�g�MHCy.g

�ŒM�/D hg��y.T�

g /;MHCy.ŒM�/i: (5-22)

460 JORG SCHURMANN

In particular,

�y.Z0/D hg��y.T

�

g /;Cy.Z/i and I�y.Z0/D hg��y.T

�

g /; ICy.Z/i:

The result of this corollary can also be seen form a differentviewpoint, by mak-ing the “going up and down” calculation already on the level of Grothendieckgroups of mixed Hodge modules, where this time one only needsthe assumptionthatf WZ0!Z is proper (to get the projection formula):

f�f�ŒM�D Œf�f

�M�D Œf�.Q

HZ 0˝f

�M/�D Œf�Q

HZ 0 �˝ŒM�2K0.MHM.Z//

for M 2 DbMHM.Z/. The problem for a singularZ is then that we do nothave a precise relation between

Œf�QHZ 0 � 2K0.MHM.Z// and ŒRf�QZ 0 � 2K0.FmHsp.Z//:

REMARK 5.13. What is missing up to now is the right notion of a good variation(or family) of mixed Hodge structures on asingularcomplex algebraic varietyZ! This class should contain at least

(1) the higher direct image local systemsRif�QZ 0 (i 2 Z) for a smooth andproper morphismf WZ0!Z of complex algebraic varieties, and

(2) the pullbackg�L of a good variation of mixed Hodge structuresL on asmooth complex algebraic manifoldM under an algebraic morphismg WZ!M .

At the moment we have to assume thatZ is smooth (and pure-dimensional), soas to use Theorem 4.3.

Nevertheless, in case (2) above we can already prove the following interestingresult (compare with [35,~ 4.1] for a similar result forMHTy� in the case whenf is a closed embedding):

COROLLARY 5.14 (MULTIPLICATIVITY ). Let f W Z ! N be a morphism ofcomplex algebraic varieties, with N smooth and puren-dimensional. Then onehas a natural pairing

f �. � /\ . � / WK0.VmHsg.N //�K0.MHM.Z//!K0.MHM.Z//;

.ŒL�; ŒM�/‘ Œf �.LH /˝M�:

HereLH Œm� is the smooth mixed Hodge module onN with underlying perversesheafLŒm�. One also has a similar pairing on(co)homological level:

f �. � /\ . � / WK0alg.N /˝ZŒy˙1��G0.Z/˝ZŒy˙1�! G0.Z/˝ZŒy˙1�;

.ŒV � �yi ; ŒF � �yj /‘ Œf �.V/˝F � �yiCj :


And the motivic Chern class transformations MHCy and MHCy commute withthese natural pairings:

MHCy

�

Œf �.LH /˝M��

DMHCy.Œf �L�/\MHCy.ŒM�/

D f ��

MHCy.ŒL�/�

\MHCy.ŒM�/ (5-23)

for L 2 VmHsg.N / andM 2DbMHM.Z/.

For the proof we can once more assumeM D g�j�V for g WM ! Z proper,with M a pure-dimensional smooth complex algebraic manifold,j WM !M

a Zariski open inclusion with complementD a normal crossing divisor withsmooth irreducible components, and finallyV a good variation of mixed Hodgestructures onM . Using the projection formula, it is then enough to prove

MHCy

�

Œg�f �.LH /˝ j�V��

DMHCy�

Œg�f �L��

\MHCy.Œj�V�/:

But g�f �L is a good variation of mixed Hodge structures onM . Therefore, byExample 5.8 and Corollary 3.14(3), both sides are equal to

�

MHCy.g�f �L/˝MHCy.j�V/

�

\�

�y

�

˝1M.log.D//

�

\ ŒOM ��

:

As an application of the very special case wheref D id W Z ! N is theidentity of a complex algebraic manifoldZ, with

MHCy.ŒQHZ �/D �y.T

�Z/\ ŒOZ �;

one gets the Atiyah–Meyer type formula (5-16) as well as the following result(cf. [12; 13; 35]):

EXAMPLE 5.15 (ATIYAH TYPE FORMULA). Let g WZ0!Z be a proper mor-phism of complex algebraic varieties, withZ smooth and connected. Assumethat for a givenM 2DbMHM.Z0/ all direct image sheaves

Rig� rat.M/ .i 2 Z/ are locally constantW

for instance,g may be a locally trivial fibration andMD QHZ 0 or MD IC H

Z 0

(for Z0 pure-dimensional), so that they all underlie a good variation of mixedHodge structures. Then one can define

ŒRg� rat.M/� WDX

i2Z

.�1/i � ŒRig� rat.M/� 2K0.VmHsg.Z//;

with

g�MHCy.ŒM�/DMHCy.g�ŒM�/

DMHCy.ŒRg� rat.M/�/˝�

�y.T�Z/\ ŒOZ �

�

: (5-24)

Here is a final application:

462 JORG SCHURMANN

EXAMPLE 5.16 (FORMULA OF ATIYAH –MEYER TYPE FOR INTERSECTION

COHOMOLOGY). Letf WZ!N be a morphism of complex algebraic varieties,with N smooth and puren-dimensional (e.g., a closed embedding). Assume alsoZ is purem-dimensional. Then one has for a good variation of mixed HodgestructuresL on N the equality

IC HZ .f �

L/Œ�m�' f �L

H ˝ IC HZ Œ�m� 2MHM.Z/Œ�m��DbMHM.Z/;

so that

ICy.ZI f�L/DMHCy.f �

L/\ICy.Z/Df��

MHCy.L/�

\ICy.Z/: (5-25)

If in addition Z is also compact, then one gets by pushing down to a point:

I�y.ZI f�L/D hMHCy.f �

L/; ICy.Z/i: (5-26)

REMARK 5.17. This example should be seen as a Hodge-theoretical version ofthe corresponding result of Banagl, Cappell, and Shaneson [4] for theL-classesL�.ICZ .L// of a selfdualPoincare local systemL on all of Z. The specialcase of Example 5.16 forf a closed inclusion was already explained in [35,~ 4.1].

Finally note that all the results of this section can easily be applied to the(un)normalizedmotivic Hirzebruch class transformation MHTy� (andAMHTy�),because theTodd class transformationtd� W G0. � / ! H�. � /˝ Q of Baum,Fulton, and MacPherson [5] has the following properties (compare also with[22, Chapter 18] and [24, Part II]):

FUNCTORIALITY: The Todd class transformationtd� commutes with push-downf� for a proper morphismf WZ! X :

td� .f� .ŒF �//D f� .td� .ŒF �// for ŒF � 2G0.Z/.

MULTIPLICATIVITY FOR EXTERIOR PRODUCTS: The Todd class transforma-tion td� commutes with exterior products:

td�

�

ŒF ˆF0��

D td� .ŒF �/ˆ td�

�

ŒF 0��

for ŒF � 2G0.Z/ andŒF 0� 2G0.Z0/.

VRR FOR SMOOTH PULLBACKS: For a smooth morphismg W Z0 ! Z ofcomplex algebraic varieties one has for the Todd class transformation td�

the following Verdier Riemann–Roch formula:

td�.Tg/\g�td�.ŒF �/D td�.g�ŒF �/D td�.Œg

�F �/ for ŒF � 2G0.Z/.

MULTIPLICATIVITY : Let ch� W K0alg. � /! H �. � /˝Q be the cohomological

Chern characterto the cohomologyH �. � / given by the operational Chow


ring CH �. � / or the usual cohomologyH 2�. � ;Z/ in even degrees. Then onehas the multiplicativity relation

td�.ŒV˝F �/D ch�.ŒV �/\ td�.ŒF �/

for ŒV � 2 K0alg.Z/ and ŒF � 2 G0.Z/, with Z a (possible singular) complex

algebraic variety.

5C. Characteristic classes and duality. In this final section we explain thecharacteristic class version of the duality formula (2-14)for the�y-genus. Wealso show that the specialization ofMHTy� for yD�1 exists and is equal to therationalized MacPherson Chern classc� of the underlying constructible sheafcomplex. The starting point is the following result [39,~ 2.4.4]:

THEOREM 5.18 (M. SAITO). Let M be a purem-dimensional complex alge-braic manifold. Then one has forM 2 DbMHM.M / the duality result(forj 2 Z)

GrFj .DR.DM//'D

�

GrF�j DR.M/

�

2Dbcoh.M /: (5-27)

HereD on the left side is the duality of mixed Hodge modules, wheresD on theright is the Grothendieck duality

DD Rhom. � ; !M Œm�/ WDbcoh.M /!Db

coh.M /;

with !M D˝mM

the canonical sheaf ofM .

A priori this is a duality for the corresponding analytic (cohomology) sheaves.SinceM andDR.M/ can be extended to smooth complex algebraic compact-ification M , one can apply Serre’s GAGA theorem to get the same result alsofor the underlying algebraic (cohomology) sheaves.

COROLLARY 5.19 (CHARACTERISTIC CLASSES AND DUALITY). Let Z bea complex algebraic variety withdualizing complex!�

Z2 Db

coh.Z/, so thatthe Grothendieck duality transformationD D Rhom. � ; !�

Z/ induces a duality

involutionD WG0.Z/!G0.Z/:

Extend this toG0.Z/˝ ZŒy˙1� by y ‘ 1=y. Then the motivic Hodge Chernclass transformation MHCy commutes with dualityD:

MHCy.D. � //DD.MHCy. � // WK0.MHM.Z//! G0.Z/˝ZŒy˙1�: (5-28)

Note that forZ D pt a point this reduces to the duality formula (2-14) forthe�y-genus. For dualizing complexes and (relative) Grothendieck duality werefer to [26; 17; 33] as well as [24, Part I,~ 7]). Note that forM smooth of puredimensionm, one has

!M Œm�' !�

M 2Dbcoh.M /:

464 JORG SCHURMANN

Moreover, for a proper morphismf W X ! Z of complex algebraic varietiesone has the relative Grothendieck duality isomorphism

Rf�

�

Rhom.F ; !�

X /�

' Rhom.Rf�F ; !�

Z / for F 2Dbcoh.X /,

so that the duality involution

D WG0.Z/˝ZŒy˙1�!G0.Z/˝ZŒy˙1�

commutes with proper push down. SinceK0.MHM.Z// is generated by classesf�ŒM�, with f WM ! Z proper morphism from a pure dimensional complexalgebraic manifoldM (andM2MHM.M /), it is enough to prove (5-28) in thecaseZ DM a pure dimensional complex algebraic manifold, in which case itdirectly follows from Saito’s result (5-27).

For a systematic study of the behavior of the Grothendieck duality trans-formationD W G0.Z/! G0.Z/ with respect to exterior products and smoothpullback, we refer to [23] and [24, Part I,~ 7], where a corresponding “bivariant”result is stated. Here we only point out that the dualities. � /_ andD commutewith thepairingsof Corollary 5.14:

f ��

. � /_�

\ .D. � //DD�

f �. � /\ . � /�

W

K0alg.N /˝ZŒy˙1��G0.Z/˝ZŒy˙1�! G0.Z/˝ZŒy˙1�;

(5-29)

and similarly

f ��

. � /_�

\ .D. � //DD�

f �. � /\ . � /�

W

K0.VmHsg.N //�K0.MHM.Z//!K0.MHM.Z//:(5-30)

Here the last equality needs only be checked for classesŒICS .L/�, with S�Z

irreducible of dimensiond andL a good variation of pure Hodge structures ona Zariski dense open smooth subsetU of S , andV a good variation of pureHodge structures onN . But then the claim follows from

f �.V/˝ ICS .L/' ICS .f�.V/jU ˝L/

and (4-3) in the form

D�

ICS .f�.V/jU ˝L/

�

' ICS

�

.f �.V/jU ˝L/_�

.d/

' ICS

�

f �.V_/jU ˝L_�

.d/:

REMARK 5.20. The Todd class transformationtd� W G0. � / ! H�. � /˝ Q,too, commutes with duality (compare with [22, Example 18.3.19] and [24, PartI, Corollary 7.2.3]) if the duality involutionD W H�. � /˝Q ! H�. � /˝Q inhomology is defined asD WD .�1/i � id onHi. � /˝Q. So also the unnormalizedHirzebruch class transformationAMHTy� commutes with duality, if this dualityin homology is extended toH�. � /˝QŒy˙1� by y‘ 1=y.


As a final result of this paper, we have:

PROPOSITION5.21.LetZ be a complex algebraic variety, and considerŒM�2

K0.MHM.Z//. Then

MHTy�.ŒM�/ 2H�.Z/˝QŒy˙1��H�.Z/˝QŒy˙1; .1Cy/�1�;

so that the specialization MHT�1�.ŒM�/ 2H�.Z/˝Q for y D �1 is defined.Then

MHT�1�.ŒM�/D c�.Œrat.M/�/DW c�.�stalk.Œrat.M/�// 2H�.Z/˝Q (5-31)

is the rationalized MacPherson Chern class of the underlying constructiblesheaf complexrat.M/ (or the constructible function�stalk.Œrat.M/�/). In par-ticular

MHT�1�.DŒM�/DMHT�1�.ŒDM�/DMHT�1�.ŒM�/: (5-32)

Here�stalk is the transformation form the diagram (4-13). Similarly, all the trans-formations from this diagram (4-13), like�stalk and rat, commute with dualityD.This implies already the last claim, sinceDD id for algebraically constructiblefunctions (compare [47,~ 6.0.6]). So we only need to prove the first part of theproposition. SinceMHT�1� andc� both commute with proper push down, wecan assumeŒM� D Œj�V�, with Z D M a smooth pure-dimensional complexalgebraic manifold,j WM !M a Zariski open inclusion with complementD

a normal crossing divisor with smooth irreducible components, andV a goodvariation of mixed Hodge structures onM . So

AMHTy�.Œj�V�/D ch��

MHCy.Rj�L/�

\AMHTy�.Œj�QHM �/2H�.M /˝QŒy˙1�

by (5-15) and themultiplicativityof the Todd class transformationtd�. Introducethe twisted Chern character

ch.1Cy/ WK0alg. � /˝QŒy˙1�!H �. � /˝QŒy˙1�;

ŒV � �yj ‘X

i�0

chi.ŒV �/ � .1Cy/i �yj ; (5-33)

with chi.ŒV �/ 2H i. � /˝Q the i-th component ofch�. Then one easily gets

MHTy�.Œj�V�/D ch.1Cy/�

MHCy.Rj�L/�

\MHTy�.Œj�QHM �/

2H�.M /˝QŒy˙1; .1Cy/�1�:

But Œj�QHM �D �Hdg.j�ŒidM �/ is by (4-15) in the image of

�Hdg WM0.var=M /DK0.var=M /ŒL�1�!K0.MHM.M //:

So forMHTy�.Œj�QHM �/ we can apply the following special case of Proposition

5.21:

466 JORG SCHURMANN

LEMMA 5.22.The transformation

Ty� DMHTy� ı�Hdg WM0.var=Z/!H�.Z/˝QŒy˙1; .1Cy/�1�

takes values inH�.Z/˝QŒy˙1��H�.Z/˝QŒy˙1; .1Cy/�1�, with

T�1� D T�1� ıD D c� ı canWM0.var=Z/!H�.Z/˝Q:

Assuming this lemma, we can derive from the following commutative diagramthat the specializationMHT�1�.Œj�V�/ for y D�1 exists:

H �. � /˝QŒy˙1��H�. � /˝QŒy˙1; .1Cy/�1�\

��! H�. � /˝QŒy˙1; .1Cy/�1�

incl:

x

?

?

?

x

?

?

?

incl:

H �. � /˝QŒy˙1��H�. � /˝QŒy˙1�\

��! H�. � /˝QŒy˙1�

yD�1

?

?

?

y

?

?

?

y

yD�1

H �. � /˝Q �H�. � /˝Q\

��! H�. � /˝Q:

Moreoverch.1Cy/ .MHCy.Rj�L// specializes fory D�1 just to

rk.L/D ch0.Œ L �/ 2H 0.M /˝Q;

with rk.L/ the rank of the local systemL on M . So we get

MHT�1�.Œj�V�/D rk.L/ � c�.j�1M /D c�.rk.L/ � j�1M / 2H�.M /˝Q;

with rk.L/ � j�1M D �stalk.rat.Œj�V�//.

It remains to prove Lemma 5.22. But all transformations —Ty�, D, c� andcan — commute with pushdown for proper maps. Moreover, by resolution ofsingularities and additivity,M0.var=Z/ is generated by classesŒf WN!Z��Lk

(k 2 Z), with N smooth puren-dimensional andf proper. So it is enough toprove thatTy�.ŒidN � �L

k/ 2H�.N /˝QŒy˙1�, with

Ty�.ŒidN � �Lk/D Ty�

�

D.ŒidN � �Lk/�

D c�

�

can.ŒidN � �Lk/�

:

But by thenormalization conditionfor our characteristic class transforma-tions one has (compare [9]):

Ty�.ŒidN �/D T �

y .TN /\ ŒN � 2H�.N /˝QŒy�;

with T�1�.ŒidN �/D c�.TN /\ ŒN �D c�.1N /. Similarly

Ty�.ŒL�/D �y.ŒQ.�1/�/D�y and can.ŒL�/D 1pt ;

so the multiplicativity ofTy� for exterior products (with a point space) yields

Ty�.ŒidN � �Lk/ 2H�.N /˝QŒy˙1�:


Moreover

T�1�.ŒidN � �Lk/D c�.1N /D c�

�

can.ŒidN � �Lk/�

:

Finally D.ŒidN � �Lk/D ŒidN � �L

k�n by the definition ofD, so that

T�1�.ŒidN � �Lk/D T�1�

�

D.ŒidN � �Lk/�

:

Acknowledgements

This paper is an extended version of an expository talk givenat the workshop“Topology of Stratified Spaces” at MSRI in September 2008. I thank the orga-nizers (G. Friedman, E. Hunsicker, A. Libgober and L. Maxim)for the invitationto this workshop. I also would like to thank S. Cappell, L. Maxim and S. Yokurafor some discussions on the subject of this paper.

References

[1] M. F. Atiyah, The signature of fiber bundles, pp. 73–84 inGlobal analysis: papersin honor of K. Kodaira, Univ. Tokyo Press, Tokyo, 1969.

[2] M. Banagl,Topological invariants of stratified spaces, Springer, Berlin, 2007.

[3] M. Banagl, The signature of singular spaces and its refinements to generalizedhomology theories, pp. 227–263 in this volume.

[4] M. Banagl, S. E. Cappell, J. L. Shaneson,Computing twisted signatures andL-classes of stratified spaces, Math. Ann. 326 (2003), 589–623.

[5] P. Baum, W. Fulton, R. MacPherson,Riemann–Roch for singular varieties, Publ.IHES 45 (1975), 101–145.

[6] P. Baum, W. Fulton, R. MacPherson,Riemann–Roch and topologicalK-theory forsingular varieties, Acta Math. 143 (1979), 155–192.

[7] A. A. Beilinson, J. Bernstein, P. Deligne,Faisceaux pervers, Asterisque 100, Soc.Math. de France, Paris, 1982.

[8] F. Bittner, The universal Euler characteristic for varieties of characteristic zero,Comp. Math. 140 (2004), 1011–1032.

[9] J. P. Brasselet, J. Schurmann, S. Yokura,Hirzebruch classes and motivic Chernclasses of singular spaces, J. Topol. Anal. 2 (2010), 1–55.

[10] S. E. Cappell, L. G. Maxim, J. L. Shaneson,Euler characteristics of algebraicvarieties, Comm. Pure Appl. Math. 61 (2008), 409–421.

[11] S. E. Cappell, L. G. Maxim, J. L. Shaneson,Hodge genera of algebraic varieties,I, Comm. Pure Appl. Math. 61 (2008), 422–449.

[12] S. E. Cappell, A. Libgober, L. G. Maxim, J. L. Shaneson,Hodge genera ofalgebraic varieties, II, Math. Annalen 345 (2009), 925–972.

468 JORG SCHURMANN

[13] S. E. Cappell, A. Libgober, L. G. Maxim, J. L. Shaneson,Hodge genera andcharacteristic classes of complex algebraic varieties, Electron. Res. Announc. Math.Sci. 15 (2008), 1–7.

[14] S. E. Cappell, J. L. Shaneson,Stratifiable maps and topological invariants, J.Amer. Math. Soc. 4 (1991), 521–551

[15] E. Cattani, A. Kaplan, W. Schmid,L2 and intersection cohomologies for apolarizable variation of Hodge structure, Inv. Math. 87 (1987), 217–252.

[16] S. S. Chern, F. Hirzebruch, J.-P. Serre,On the index of a fibered manifold, Proc.Amer. Math. Soc. 8 (1957), 587–596.

[17] B. Conrad,Grothendieck Duality and Base Change, Lecture Notes in Mathemat-ics, Vol. 1750. Springer, 2000.

[18] P. Deligne,Theoreme de Lefschetz et criteres de degenerescence de suites spec-trales, Publ. Math. IHES 35 (1968), 107–126.

[19] P. Deligne,Equation differentielles a point singular regulier, Springer, Berlin,1969.

[20] P. Deligne,Theorie de Hodge II, Publ. Math. IHES 40 (1971), 5–58.

[21] P. Deligne,Theorie de Hodge III, Publ. Math. IHES 44 (1974), 5–78.

[22] W. Fulton,Intersection theory, Springer, 1981.

[23] W. Fulton, L. Lang,Riemann–Roch algebra, Springer, New York, (1985).

[24] W. Fulton, R. MacPherson,Categorical framework for the study of singularspaces, Memoirs Amer. Math. Soc. 243 (1981).

[25] M. Goresky, R. MacPherson,Intersection homology II, Invent. Math. 71 (1983),77–129.

[26] R. Hartshorne,Residues and duality, Lecture Notes in Mathematics, Vol. 20.Springer, New York, 1966.

[27] F. Hirzebruch,Topological methods in algebraic geometry, Springer, Berlin, 1966.

[28] M. Kashiwara,A study of a variation of mixed Hodge structures, Publ. RIMS 22(1986), 991–1024.

[29] M. Kashiwara, T. Kawai,The Poincare lemma for variations of polarized Hodgestructures, Publ. RIMS 23 (1987), 345–407.

[30] M. Kashiwara, P. Schapira,Sheaves on manifolds, Springer, Berlin, 1990.

[31] G. Kennedy,MacPherson’s Chern classes of singular varieties, Comm. Alg. 18(1990), 2821–2839.

[32] A. Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms,pp. 99–125 in these proceedings.

[33] J. Lipmann, M. Hashimoto,Foundations of Grothendieck Duality for Diagrams ofSchemes, Lecture Notes in Mathematics, Vol. 1960. Springer, 2009.

[34] R. MacPherson,Chern classes for singular algebraic varieties, Ann. of Math. (2)100 (1974), 423–432.


[35] L. Maxim, J. Schurmann,Hodge-theoretic Atiyah–Meyer formulae and the strat-ified multiplicative property, pp. 145–167 in “Singularities I: algebraic and analyticaspects”, Contemp. Math. 474 (2008).

[36] W. Meyer, Die Signatur von lokalen Koeffizientensystemen und Faserbundeln,Bonner Mathematische Schriften 53, (Universitat Bonn), 1972.

[37] C. Peters,Tata lectures on motivic aspects of Hodge theory, Lecture Notes of theTata Institute of Fundamental Research, 2010.

[38] C. Peters, J. Steenbrink,Mixed Hodge structures, Ergebnisse der Math. (3) Vol. 52,Springer, Berlin, 2008.

[39] M. Saito,Modules de Hodge polarisables, Publ. RIMS 24 (1988), 849–995.

[40] M. Saito,Mixed Hodge modules, Publ. RIMS 26 (1990), 221–333.

[41] M. Saito,Introduction to mixed Hodge modules, pp. 145–162 in Actes du Colloquede Theorie de Hodge (Luminy, 1987), Asterisque Vol. 179–180 (1989).

[42] M. Saito,On Kollar’s Conjecture, Proceedings of Symposia in Pure Mathematics52, Part 2 (1991), 509–517.

[43] M. Saito,On the formalism of mixed sheaves, preprint, arXiv:math/0611597.

[44] M. Saito,Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000),283–331.

[45] C. Sabbah,Hodge theory, singularities andD-modules, preprint (2007), home-page of the author.

[46] W. Schmid,Variation of Hodge structures: the singularities of the period mapping,Inv. Math. 22 (1973), 211–319.

[47] J. Schurmann,Topology of singular spaces and constructible sheaves, MonografieMatematyczne, Vol.63, Birkhauser, Basel, 2003.

[48] J. Schurmann, S. Yokura,A survey of characteristic classes of singular spaces,pp. 865–952 in “Singularity theory” (Marseille, 2005), edited by D. Cheniot et al.,World Scientific, Singapore, 2007.

[49] P. H. Siegel,Witt spaces: A geometric cycle theory forKO-homology at oddprimes, Amer. J. Math. 105 (1983), 1067–1105.

[50] J. Steenbrink, S. Zucker,Variations of mixed Hodge structures, Inv. Math. 80(1983).

[51] B. Totaro,Chern numbers for singular varieties and elliptic homology, Ann. ofMath. (2) 151 (2000), 757–791.

[52] C. Voisin,Hodge theory and complex algebraic geometry I, Cambridge Studies inAdvanced Mathematics 76, Cambridge University Press, 2002.

[53] R. Waelder,Rigidity of differential operators and Chern numbers of singularspaces, pp. 35–54 in this volume.

[54] J. Woolf,Witt groups of sheaves on topological spaces, Comment. Math. Helv. 83(2008), 289–326.

470 JORG SCHURMANN

[55] S. Yokura,Motivic characteristic classes, pp. 375–418 in these proceedings.

[56] S. Zucker,Hodge theory with degenerating coefficients:L2-cohomology in thePoincare metric, Ann. Math. 109 (1979), 415–476.

JORG SCHURMANN

MATHEMATISCHE INSTITUT

UNIVERSITAT M UNSTER

EINSTEINSTR. 6248149 MUNSTER

GERMANY

[email protected]


Workshop on the Topology of Stratified SpacesOpen Problems

The following open problems were suggested by the participants both duringand following the Workshop.

1. L2 Hodge and signature theorems;

Signature theory on singular spaces

(a) (suggested by Eugenie Hunsicker)Consider a pseudomanifoldX as in Cheeger, [7]. Cheeger proves in this

paper that if the smooth part ofX , Xreg, is endowed with an iterated conemetric, and ifX is a Witt space, then theL2 cohomology ofXreg is iso-morphic to the middle perversity intersection cohomology of X (which isalso unique due to the Witt condition). This implies in turn that the space ofL2 harmonic forms for the maximal extension is isomorphic to the middleperversity intersection cohomology, and from this we get that the operatord Cı is essentially self-adjoint onXreg, which in turn means that this operatorhas a unique closed extension toL2.Xreg/: Thus in the setting of Witt spacesand conical metrics, there is a clear and simple relationship between harmonicforms,L2-cohomology and intersection cohomology.

If the Witt condition is dropped, then there is not generallya unique mid-dle perversity intersection cohomology, andd C ı generally has differentpossible extensions, and in particular, can have differentpossible self-adjointextensions. In the case of a pseudomanifold with only one singular stratumendowed again with a conical metric, the non-Witt case was studied in [14].In this paper, it is shown that the operatord Cı onXreg endowed with a conemetric has self-adjoint extensions whose kernels are isomorphic to the upperand to the lower middle perversity intersection cohomologies onX , and thekernel of the minimal extension ofd C ı (for an appropriately chosen conemetric) is isomorphic to the image of lower middle in upper middle perversityintersection cohomology.

471

472 OPEN PROBLEMS

Versions ofL2 cohomology also can relate to more general perversities.For example, Nagase showed in [18] that for each standard perversity p

greater than or equal to the upper middle perversity on a pseudomanifoldX , there exists an incomplete metric on the regular set ofX for which theL2 cohomology associated to the maximal extension ofd is isomorphic toIH p.X /. This in particular implies that there exists a self-dualL2 extensionof d C ı for this metric whose kernel is also isomorphic toIH p.X /.

It seems likely that these phenomena are part of a larger relationship amongL2 extensions of the geometric operatord C ı on the regular set of a pseu-domanifold for various metrics, weightedL2 cohomology for these metricsand intersection cohomologies onX with various perversities. It would beinteresting to explore this further. In particular, consider the metrics con-structed in [18]. What other closedL2 extensions ofd C ı exist for thesemetrics, and which perversity intersection cohomologies will their kernels beisomorphic to? Further, are there generalizations of intersection cohomologythat are isomorphic to the kernel of some such extensions? Finally, can weunderstand the interesting extensions ofd C ı using analytic approaches tosingularities, such as the Melroseb-calculus or Schulze or Boutet de Monvelcalculi?

(b) (suggested by Paolo Piazza)Consider a non-Witt pseudomanifoldX that has a Lagrangian structurea

la Banagl. See for example Chapter 9 in the book [4]. For such spaces onecan define a signature and an L-class.

(i) Is there an analytic description of the signature? More precisely: endowX with an iterated conic metric. Is there an extension of the signatureoperator which is Fredholm and such that its index is equal tothe abovesignature?

For Witt spaces this is a well known result due to Cheeger and recentlyre-established by Albin, Leichtnam, Mazzeo and Piazza in the preprint[1]. In the latter preprint the extension of the full signature package fromclosed manifolds to Witt spaces, leading to the definition and the homo-topy invariance of higher signatures on Witt spaces, is discussed. Noticethat the higher signatures on Witt spaces involve the L-class of Goresky–MacPherson.

(ii) What part of the signature package on closed manifolds and Witt spacescan be extended to these non-Witt spaces?

(c) (suggested by Shmuel Weinberger)What kind of elliptic operator theory “tracks” (i.e. contains a signature

operator forG-manifolds) the cosheaf homology term in stratified surgery

OPEN PROBLEMS 473

for M=G? Things are easy whenG acts locally freely so the quotient is anorbifold but this looks interesting in general.

(d) (suggested by Shmuel Weinberger; clarifications by Les Saper and PaoloPiazza)

The entrance of sheaves with nontrivial local cohomology whose globalvanishing is important for global self-duality in Saper’s talk suggests thatcompactifications have global “index invariants” inL.RP / or K.CP / butdo not localize, i.e. pullback toK.B� / (when� has torsion). For� withtorsion, a “pullback” probably would be accidental nonsense. This is likewhat happens (for a different reason) in Fowler’s talk on uniform latticeswith torsion.

(e) (suggested by David Trotman)If P andP 0 are homeomorphic PL Witt spaces (i.e. you think of them as

different triangulations of the same object), it is a consequence of Goresky–MacPherson II and Siegel (or other combinations) that theseare cobordant inthe Witt sense. How elementary is this fact? Is there adirect proof?

2. Topology of algebraic varieties

(a) (suggested by Anatoly Libgober)Does there exist a cobordism theory of pairs.X;D/ such that forD log-

terminal,E l l.X;D/ is invariant under such cobordisms? See [6] for a dis-cussion of elliptic genus of pairs and results related to this question.

(b) (suggested by Clint McCrory)

(i) Define intersection homology for real algebraic varieties. This questionappears on Goresky and MacPherson’s 1994 problem list [9]. Interestingwork has been done by van Hamel [21]; see also [3; 20].

(ii) Simplify Akbulut and King’s conjectural topological characterization ofreal algebraic varieties [2], and compute the bordism ring of real algebraicvarieties. Invariants are “Akbulut–King numbers” [15].

(iii) What is the topology of the weight filtration of a real algebraic variety[17]? How can the filtration vary within a homeomorphism type? Is thefiltration trivial for Z2 homology manifolds? Is it a bi-Lipshitz invariant(Trotman)?

(iv) Prove that the Stiefel–Whitney homology classes of a real algebraic va-riety are topologically invariant (cf. [8]).

(v) Which real toric varietiesX are maximal, that is, when is

dimH�.X.R/I Z2/D dimH�.X.C/I Z2/?

See Hower’s counterexample [12].

474 OPEN PROBLEMS

(vi) If a complex algebraic variety is defined overR, what is the relationbetween the Deligne weight filtration of the cohomology (or homology)of the complex points and the weight filtration defined by Totaro [20] andMcCrory and Parusinski [17] for the real points? The weight filtration of thehomology of a complex variety can be defined with arbitrary coefficients.What is the relation between the weight filtration of the homology of thecomplex points withZ2 coefficients and the weight filtration for the realpoints?

(vii) Are there motivic characteristic classes for real varieties analogous tothose defined by Brasselet, Schurmann, and Yokura [22] for complex vari-eties? The virtual Betti numbersq of real algebraic varieties [16] satisfythe “scissor relations”

ˇq.X /D ˇq.Y /Cˇq.X n Y /

for Y a closed subvariety ofX . Can the virtual Betti numbers be extendedto characteristic classes of real varieties?

(c) (suggested by David Trotman)

(i) Is it true that every topologically conical complex stratification of a com-plex analytic variety is Whitney.A/-regular? (This is not true for realalgebraic varieties.)

(ii) Does every WhitneyC k stratified set admit aC k triangulation such thatthe open simplices are strata of a Whitney stratification? The same questionreplacing “Whitney” by “Bekka.”

(iii) It is known that families of (germs of) complex hypersurfaces with anisolated singularity have constant Milnor number if an onlyif they haveconstant topological type (except for “only if” for surfaces where it is anopen question). Could it be true that having constant topological type isequivalent to the family being BekkaC -regular over the parameter space?

(iv) Can Goresky–MacPherson’s Morse theory be made to work for tameBekka stratifications instead of tame Whitney stratifications?

(v) It is known (Noirel -1996) that every abstract stratifiedspace (Thom–Mather space) can be embedded in someR

n as a semi-algebraic Whitneystratified set (even Verdier regular) with semi-algebriac control data withoutrefining the original stratification. Is there such a Mostowski stratified em-bedding, or at least locally bi-Lipschitz trivial semi-algebraic stratificationwithout refinement? (By theorems of Parusinski (1992) or Valette (2005),there are refinements with these properties.)

(vi) Suppose 2 germs of complex analytic functions onCn with isolated

singularities at0 are topologically equivalent, i.e. there exists a homeo-

OPEN PROBLEMS 475

morphismh W .Cn; 0/! .Cn; 0/ overf W .Cn; 0/! C andg W .Cn; 0/! C

such thatf D gh. Can one find another such homeomorphismh1 suchthat h1 preserves distance to the origin, i.e.kh1.z/k D kzk for z near0 2 C

n? (A positive answer would solve Zariski’s 1970 problem about thetopological invariance of the multiplicity.)

3. Mixed Hodge theory and singularities

(suggested by Matt Kerr and Gregory Pearlstein)

The period domain classifying Hodge structures of type.hn;0; hn�1;1; : : : ; h0;n/,(say all> 0), n > 1 odd, is anon-locally-symmetric homogeneous space. Un-derstand theL2-cohomology groups

Hq

.2/

�

� nD;˝k..�hk;n�k

/Hk;n�k/˝ak�

and the role played by these in algebraizing images of periodmaps. (Note that� is an arithmetic subgroup andK� nD is a line bundle of the form shown.)Possible reference (somewhat outdated): [11].

4. Characteristic class theories for singular varieties

(a) (suggested by Jorg Schurmann)We work in the algebraic context overC. Find a pure-dimensional variety

X such that the class of the intersection cohomology complexICX in theGrothendieck group of complex algebraically constructible sheaves

ŒICX � 2 K0.Dbc .X //

is not in the subgroup generated byŒRf�QZ � with Z smooth pure-dimen-sional andf W Z ! X proper.

Note that it is important to take only the classes of the totaldirect images.If one asks the same question for the subgroup generated bydirect summandsof ŒRf�QZ � with Z smooth pure-dimensional andf W Z ! X proper, thenŒICX � belongs to this subgroup by thedecomposition theorem(compare [19,Corollary 4.6], for example).

A positive answer to the question above, stated in the “topological context”of algebraically constructible sheaves, would also give anexample such thatthe class

ŒIC HX � 2 K0.MHM.X //

of the corresponding (pure) intersection Hodge moduleIC HX

in the Grothen-dieck group of algebraic mixed Hodge modules isnot in the image of the

476 OPEN PROBLEMS

natural group homomorphism

�Hdg W K0.var=X /! K0.MHM.X //

from themotivic relative Grothendieck groupof complex algebraic varietiesoverX (compare [19, Section 4.2]).

This fact would further justify the study of characteristicclasses of mixedHodge modules in the works of Cappell, Libgober, Maxim, Schurmann, andShaneson; see [19] and the references therein.

(b) (suggested by Shmuel Weinberger)Are the elliptic genera, etc. part of an integral theory the way L-classes

come back fromKO.M / in index theory? Schurmann and Yokura knowsomething about this but with too few variables.

References

[1] Pierre Albin, Eric Leichtnam, Rafe Mazzeo, Paolo Piazza, The signature packageon Witt spaces, I. Index classes, http://arxiv.org/abs/0906.1568

[2] S. Akbulut, H. King, Topology of real algebraic sets, MSRI Publ.25, SpringerVerlag, New York, 1992.

[3] M. Banagl, The signature of singular spaces and its refinements to generalizedhomology theories, in this volume.

[4] M. Banagl, Topological invariants of stratified spaces, Springer Monographs inMathematics, Springer, New York, 2006

[5] Jean-Paul Brasselet, Joerg Schurmann, Shoji Yokura,Hirzebruch classes and mo-tivic Chern classes for singular spaces, J. Topol. Anal. 2:1 (2010), 1–55.

[6] Lev Borisov, Anatoly Libgober,Elliptic genera of singular varieties, Duke Math. J.116 (2003), 319–351.

[7] Cheeger, Jeff,On the Hodge theory of Riemannian pseudomanifolds. Geometry ofthe Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii,1979), pp. 91–146, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., 1980.

[8] J. Fu, C. McCrory,Stiefel–Whitney classes and the conormal cycle of a real analyticvariety, Trans. Amer. Math. Soc.349(1997), 809–835.

[9] M. Goresky, R. MacPherson,Problems and bibliography on intersection homology,in Intersection cohomology, A. Borelet al., Birkhauser 1994, 221–229.

[10] P. Griffiths, M. Green, M. Kerr,Some enumerative global properties of variationsof Hodge structure, to appear in Moscow Math. J.

[11] P. Griffiths, W. Schmid,Locally homogeneous complex manifolds, Acta Math.123(1969), 253-302.

[12] V. Hower, A counterexample to the maximality of toric varieties, Proc. Amer.Math. Soc.,136(2008), 4139–4142.

OPEN PROBLEMS 477

[13] E. Hunsicker,Hodge and signature theorems for a family of manifolds with fibrebundle boundary, Geom. Topol. 11 (2007), 1581–1622.

[14] E. Hunsicker, R. Mazzeo,Harmonic forms on manifolds with edges, Int. Math.Res. Not. (2005) .

[15] C. McCrory, A. Parusinski, The topology of real algebraic sets of dimension 4:necessary conditions, Topology39 (2000), 495–523.

[16] C. McCrory, A. Parusinski, Virtual Betti numbers of real algebraic varieties,Comptes Rendus Acad. Sci. Paris, Ser. I,336(2003), 763–768.

[17] C. McCrory, A. Parusinski, The weight filtration for real algebraic varieties, inthis volume.

[18] M. Nagase,L2-cohomology and intersection homology of stratified spaces, DukeMath. J. 50 (1983).

[19] J. Schurmann,Characteristic classes of mixed Hodge modules, in this volume.

[20] B. Totaro,Topology of singular algebraic varieties, Proc. Int. Cong. Math. Beijing(2002), 533-541.

[21] J. van Hamel,Towards an intersection homology theory for real algebraicvari-eties, Int. Math. Research Notices25 (2003), 1395–1411.

[22] S. Yokura,Motivic characteristic classes, in this volume.

cohomology - maths.ed.ac.ukv1ranick/papers/topstrat.pdf · success in expanding to stratiﬁed...

Documents