hodge theory and complex algebraic...

HODGE THEORY AND COMPLEXALGEBRAIC GEOMETRY I

CLAIRE VOISINCNRS, Institut de Mathematiques de Jussieu

Translated by Leila Schneps

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C© Cambridge University Press 2002

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no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 2002

Printed in the United Kingdom at the University Press, Cambridge

TypefaceTimes 10/13 pt SystemLATEX2ε [tb]

A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication data

Voisin, Claire.Hodge theory and complex algebraic geometry / Claire Voisin.

p. cm. – (Cambridge studies in advanced mathematics)Includes bibliographical references and index.

ISBN 0 521 80260 11. Hodge theory. 2. Geometry, Algebraic. I. Title. II. Series.

QA564 .V65 2002516.3′5 – dc21 2002017389

ISBN 0 521 80260 1 hardback

Contents

0 Introduction page1I Preliminaries 191 Holomorphic Functions of Many Variables 21

1.1 Holomorphic functions of one variable 221.1.1 Definition and basic properties 221.1.2 Background on Stokes’ formula 241.1.3 Cauchy’s formula 27

1.2 Holomorphic functions of several variables 281.2.1 Cauchy’s formula and analyticity 281.2.2 Applications of Cauchy’s formula 30

1.3 The equation∂g∂z = f 35

Exercises 372 Complex Manifolds 38

2.1 Manifolds and vector bundles 392.1.1 Definitions 392.1.2 The tangent bundle 412.1.3 Complex manifolds 43

2.2 Integrability of almost complex structures 442.2.1 Tangent bundle of a complex manifold 442.2.2 The Frobenius theorem 462.2.3 The Newlander–Nirenberg theorem 50

2.3 The operators∂ and∂ 532.3.1 Definition 532.3.2 Local exactness 552.3.3 Dolbeault complex of a holomorphic bundle 57

2.4 Examples of complex manifolds 59Exercises 61

v

vi Contents

3 Kahler Metrics 633.1 Definition and basic properties 64

3.1.1 Hermitian geometry 643.1.2 Hermitian and Kähler metrics 663.1.3 Basic properties 67

3.2 Characterisations of Kähler metrics 693.2.1 Background on connections 693.2.2 Kahler metrics and connections 71

3.3 Examples of Kähler manifolds 753.3.1 Chern form of line bundles 753.3.2 Fubini–Study metric 763.3.3 Blowups 78

Exercises 824 Sheaves and Cohomology 83

4.1 Sheaves 854.1.1 Definitions, examples 854.1.2 Stalks, kernels, images 894.1.3 Resolutions 91

4.2 Functors and derived functors 954.2.1 Abelian categories 954.2.2 Injective resolutions 964.2.3 Derived functors 99

4.3 Sheaf cohomology 1024.3.1 Acyclic resolutions 1034.3.2 The de Rham theorems 1084.3.3 Interpretations of the groupH1 110

Exercises 113II The Hodge Decomposition 1155 Harmonic Forms and Cohomology 117

5.1 Laplacians 1195.1.1 TheL2 metric 1195.1.2 Formal adjoint operators 1215.1.3 Adjoints of the operators∂ 1215.1.4 Laplacians 124

5.2 Elliptic differential operators 1255.2.1 Symbols of differential operators 1255.2.2 Symbol of the Laplacian 1265.2.3 The fundamental theorem 128

5.3 Applications 1295.3.1 Cohomology and harmonic forms 129

Contents vii

5.3.2 Duality theorems 130Exercises 136

6 The Case of K¨ahler Manifolds 1376.1 The Hodge decomposition 139

6.1.1 Kahler identities 1396.1.2 Comparison of the Laplacians 1416.1.3 Other applications 142

6.2 Lefschetz decomposition 1446.2.1 Commutators 1446.2.2 Lefschetz decomposition on forms 1466.2.3 Lefschetz decomposition on the cohomology 148

6.3 The Hodge index theorem 1506.3.1 Other Hermitian identities 1506.3.2 The Hodge index theorem 152

Exercises 1547 Hodge Structures and Polarisations 156

7.1 Definitions, basic properties 1577.1.1 Hodge structure 1577.1.2 Polarisation 1607.1.3 Polarised varieties 161

7.2 Examples 1677.2.1 Projective space 1677.2.2 Hodge structures of weight 1 and abelian varieties 1687.2.3 Hodge structures of weight 2 170

7.3 Functoriality 1747.3.1 Morphisms of Hodge structures 1747.3.2 The pullback and the Gysin morphism 1767.3.3 Hodge structure of a blowup 180

Exercises 1828 Holomorphic de Rham Complexes and Spectral Sequences 184

8.1 Hypercohomology 1868.1.1 Resolutions of complexes 1868.1.2 Derived functors 1898.1.3 Composed functors 194

8.2 Holomorphic de Rham complexes 1968.2.1 Holomorphic de Rham resolutions 1968.2.2 The logarithmic case 1978.2.3 Cohomology of the logarithmic complex 198

8.3 Filtrations and spectral sequences 2008.3.1 Filtered complexes 200

viii Contents

8.3.2 Spectral sequences 2018.3.3 The Fr¨olicher spectral sequence 204

8.4 Hodge theory of open manifolds 2078.4.1 Filtrations on the logarithmic complex 2078.4.2 First terms of the spectral sequence 2088.4.3 Deligne’s theorem 213

Exercises 214III Variations of Hodge Structure 2179 Families and Deformations 219

9.1 Families of manifolds 2209.1.1 Trivialisations 2209.1.2 The Kodaira–Spencer map 223

9.2 The Gauss–Manin connection 2289.2.1 Local systems and flat connections 2289.2.2 The Cartan–Lie formula 231

9.3 The Kahler case 2329.3.1 Semicontinuity theorems 2329.3.2 The Hodge numbers are constant 2359.3.3 Stability of Kahler manifolds 236

10 Variations of Hodge Structure 23910.1 Period domain and period map 240

10.1.1 Grassmannians 24010.1.2 The period map 24310.1.3 The period domain 246

10.2 Variations of Hodge structure 24910.2.1 Hodge bundles 24910.2.2 Transversality 25010.2.3 Computation of the differential 251

10.3 Applications 25410.3.1 Curves 25410.3.2 Calabi–Yau manifolds 258

Exercises 259IV Cycles and Cycle Classes 26111 Hodge Classes 263

11.1 Cycle class 26411.1.1 Analytic subsets 26411.1.2 Cohomology class 26911.1.3 The Kahler case 27311.1.4 Other approaches 275

Contents ix

11.2 Chern classes 27611.2.1 Construction 27611.2.2 The Kahler case 279

11.3 Hodge classes 27911.3.1 Definitions and examples 27911.3.2 The Hodge conjecture 28411.3.3 Correspondences 285

Exercises 28712 Deligne–Beilinson Cohomology and the Abel–Jacobi Map 290

12.1 The Abel–Jacobi map 29112.1.1 Intermediate Jacobians 29112.1.2 The Abel–Jacobi map 29212.1.3 Picard and Albanese varieties 296

12.2 Properties 30012.2.1 Correspondences 30012.2.2 Some results 302

12.3 Deligne cohomology 30412.3.1 The Deligne complex 30412.3.2 Differential characters 30612.3.3 Cycle class 310

Exercises 313

Bibliography 315Index 319

1Holomorphic Functions of Many Variables

In this chapter,we recall themainproperties of holomorphic functionsof severalcomplex variables. These results will be used freely in the remainder of thetext, and will enable us to introduce the notions of a complex manifold, and aholomorphic function defined locally on a complex manifold.TheC-valued holomorphic functions of the complex variablesz1, . . . , zn are

thosewhose differential isC-linear, or equivalently, thosewhich are annihilatedby the operators∂

∂zi. It follows immediately from this definition that the set of

holomorphic functions forms a ring, and that the composition of two holo-morphic functions is holomorphic. The following theorem, however, requiresa certain amount of work.

Theorem 1.1 The holomorphic functions of the complex variables z1, . . . , znare complex analytic, i.e. they locally admit expansions as power series in thevariables zi .

This result is an easy consequence of Cauchy’s formula in several variables,which is a generalisation of the formula

f (z) = 1

2iπ

∫|ζ |=1

f (ζ )

ζ − zdζ,

where f is a holomorphic function defined in a disk of radius> 1, and|z| < 1.Cauchy’s formula can also be used to prove Riemann’s theorem of analytic

continuation:

Theorem 1.2 Let f be a bounded holomorphic function on the pointed disk.Then f extends to a holomorphic function on the whole disk.

And also Hartogs’ theorem:

21

22 1 Holomorphic Functions of Many Variables

Theorem 1.3 Let f be a holomorphic function defined on the complementof the subset F defined by the equations z1 = z2 = 0 in a ball B ofCn, n ≥ 2.Then f extends to a holomorphic function on B.

(More generally, this theorem remains true ifF is an analytic subset of codi-mension 2, but we only need the present version here.) Hartogs’ theorem isused more in complex geometry than Riemann’s theorem, because it does notimpose any conditions on the functionf . More generally, it enables us to showthat a holomorphic section of a complex vector bundle over a complex man-ifold is defined everywhere if it is defined on the complement of an analyticsubset of codimension 2. This is classically used to show the invariance of the“plurigenera” under birational transformations.We conclude this chapter with a proof of an explicit formula for the local

solution of the equation

∂ f

∂z= g,

whereg is a differentiable function definedonanopenset ofC. Thiswill be usedin the following chapter, to prove the local exactness of the Dolbeault complex.A good reference for the material in this chapter is H¨ormander (1979).

1.1 Holomorphic functions of one variable

1.1.1 Definition and basic properties

LetU ⊂ C ∼= R2 be an open set, andf : U → C a C1 map. Letx, y be the

linear coordinates onR2 such thatz = x + iy is the canonical linear complexcoordinate onC. Consider the complex-valued differential form

dz= dx+ idy ∈ HomR (TU , C) ∼= �U,R ⊗ C.

Clearlydzand its complex conjugatedz form a basis of�U,R ⊗ C overC atevery point ofU , since

2dx = dz+ dz, 2idy = dz− dz. (1.1)

The complex differential formd f ∈HomR (TU , C) can thus be uniquelywritten

d fu = fz(u)dz+ fz(u)dz, (1.2)

where the complex-valued functionsu �→ fz(u), u �→ fz(u) are continuous.

Definition 1.4 We write fz = ∂ f∂z and fz = ∂ f

∂z .

1.1 Holomorphic functions of one variable 23

By (1.1) we obviously have

∂ f

∂z= 1

2

(∂ f

∂x− i

∂ f

∂y

),

∂ f

∂z= 1

2

(∂ f

∂x+ i

∂ f

∂y

). (1.3)

We can also consider the decomposition (1.2) as the decomposition ofd f ∈HomR(C, C) intoC-linear andC-antilinear parts:

Lemma 1.5We have∂ f∂z (u) = 0 if and only if theR-linear map

d fu : TU,u∼= C → C

is C-linear, i.e. is equal to multiplication by a complex number, which is thenequal to∂ f

∂z (u).

Proof Because∂∂y = i ∂

∂x for the natural complex structure onTU,u, the mor-phismd fu : TU,u → C isC-linear if and only if we have

∂ f

∂y(u) = i

∂ f

∂x(u),

and by (1.3), this is equivalent to∂ f∂z (u) = 0. Moreover, we then haved fu =

∂ f∂z (u)dz, i.e.

d fu

(∂

∂x

)= ∂ f

∂z(u), d fu

(∂

∂y

)= i

∂ f

∂z(u),

which proves the second assertion, since the natural isomorphismTU,u∼= C

sends ∂∂x to 1. �

Definition 1.6 The function f is said to be holomorphic if it satisfies one of theequivalent conditions of lemma 1.5 at every point of U.

Lemma 1.7 If f is holomorphic and does not vanish on U, then1f is holomor-phic. Similarly, if f, g are holomorphic, f g and f+ g and g◦ f (when g isdefined on the image of f ) are all holomorphic.

Proof Themapz �→ 1z is holomorphic onC

∗, so that the first assertion followsfrom the last one. Furthermore, ifg and f areC1 andg is defined on the imageof f , theng ◦ f is C1 and we have

d(g ◦ f )u = dgf (u) ◦d fu.


If dgf (u) andd fu are bothC-linear for the natural identifications ofTC,u, TC, f (u)

andTC,g◦ f (u) with C, thend(g ◦ f )u is alsoC-linear, and the last assertion isproved. The other properties are proved similarly. �

In particular, we will use the following corollary.

Corollary 1.8 If f is holomorphic on U, the map g defined by

g(z) = f (z)

z− a

is holomorphic on U− {a}.

1.1.2 Background on Stokes’ formula

Letα be aC1 differentialk-form on ann-dimensionalmanifoldU (cf. definition2.3andsection2.1.2 in the followingchapter). Ifx1, . . . , xn are local coordinatesonU , we can write

α =∑I

αI dxI ,

where the indicesI parametrise the totally ordered subsetsi1 < · · · < i k of{1, . . . ,n}, with dxI = dxi1 ∧ · · · ∧ dxik . We can then define the continuous(k+ 1)-form

dα =∑I ,i

∂αI

∂xidxi ∧ dxI ; (1.4)

we check that it is independent of the choice of coordinates. This follows fromthe more general fact that ifV is anm-dimensional manifold andφ : V → Uis a C1 map given in local coordinates byφ∗xi := xi ◦ φ = φi (y1, . . . , ym),then for every differential formα = ∑

I αI dxi1 ∧ · · · ∧ dxik , we can define itsinverse image

φ∗α =∑I

αI ◦ φ dφi1 ∧ · · · ∧ dφi k .

Moreover, ifφ is C2, this image inverse satisfiesd(φ∗α) = φ∗(dα),

where the coordinatesyi (and the formulae (1.4)) are used on the left, while thecoordinatesxi are used on the right.


A C0 differential k-form α can be integrated over the compact orientedk-dimensional submanifolds ofU with boundary, or over the image of suchmanifolds under differentiable maps.To begin with, let us recall that ak-dimensional manifold with boundary

is a topological space covered by open setsUi which are homeomorphic, viacertain mapsφi , to open subsets ofRk or to ]0,1] × V , whereV is an openset ofRk−1. We require the transition functionsφi ◦ φ−1

j to be differentiable onφ j (Ui ∩Uj ). Whenφ j (Ui ∩Uj ) contains points on the boundary ofUj , i.e. islocally isomorphic to ]0,1]× V , whereV is an open set ofRk−1, φi (Ui ∩Uj )must also be locally isomorphic to ]0,1]×V ′, whereV ′ is an open set ofRk−1,and the differentiable mapφi ◦ φ−1

j must locally extend to a diffeomorphismof a neighbourhood inRk of ]0,1] × V to a neighbourhood of ]0,1] × V ′,inducing a diffeomorphism from 1×V to 1×V ′. In particular, the boundary ofM , which we denote by∂M and which is defined, with the preceding notation,as the union of theφ−1

i (1×V), is a closed set ofM which possesses an induceddifferentiable manifold structure.The manifold with boundaryM is said to be oriented if the diffeomorphisms

φi ◦ φ−1j have positive Jacobian. The boundary ofM is then also naturally

oriented by the charts 1× V , whereV is an open set ofRk−1 as above, sincethe induced transition diffeomorphisms

φi ◦ φ−1j |1×V : V → V ′

also have positive Jacobian.If M is a k-dimensional manifold with boundary andφ : M → U is aC1

differentiable map (along the boundary ofM , which is locally isomorphic to]0,1] × V , we requireφ to extend locally to aC1 map on a neighbourhood]0,1+ ε[ × V of {1} × V), then for every continuousk-form α, we have theinverse imageβ = φ∗α defined above, which is a continuousk-form onM . IfmoreoverM is oriented and compact, such a form can be integrated overM asfollows. Let fi be a partition of unity subordinate to a covering ofM by opensetsUi as above,whichwemay assume to be diffeomorphic to ]0,1]× ]0,1[k−1

or to ]0,1[k. Thenβ = ∑i fiβ, and the formfiβ onUi extends to a contin-

uous form on [0,1]k. Setting fiβ = gi (x1, . . . , xk)dx1 ∧ · · · ∧ dxk, we thendefine

∫M

β =∑i

∫Ui

fiβ

∫Ui

fiβ =∫ 1

0· · ·

∫ 1

0gi (x1, . . . , xk)dx1 . . .dxk.


The change of variables formula for multiple integrals and the fact that theauthorised variable changes have positive Jacobians ensure that

∫M β is well-

defined independently of the choice of oriented charts, i.e. of local orientation-preserving coordinates.

Remark 1.9 If we change the orientation of M, i.e. if we compose all thecharts with local diffeomorphisms ofR

k with negative Jacobians, the integrals∫M φ∗α change sign. This follows from the change of variables formula formultiple integrals, which uses only the absolute value of the Jacobian, whereasthe change of variables formula for differential forms of maximal degree usesthe Jacobian itself.

Suppose now thatα is aC1 (k−1)-form onU . Then, asφ|∂M is differentiableand∂M is a compact oriented manifold of dimensionk − 1, we can computethe integral

∫∂M φ∗α. Moreover, we can integrate the differentialφ∗dα = dφ∗α

overM . We then have

Theorem 1.10 (Stokes’ formula) The following equality holds:

∫M

φ∗dα =∫

∂Mφ∗α. (1.5)

In particular, if dα = 0, we have∫∂M φ∗α = 0.

Proof Using a partition of unity, we are reduced to showing (1.5) whenφ∗αhas compact support in an open setUi of M as above. This follows immediatelyfrom the formula (1.4) for the differential, and the equality

∫ 1

0f ′(t)dt = f (1)− f (0),

which holds for anyC1 function f . �

Wewill useStokes’ formulavery frequently throughout this text. Inparticular,it will enable us to pair the de Rham cohomology with the singular homology.The following consequence will be particularly useful.

Corollary 1.11 If α is a differential form of degree n− 1 on a compact n-dimensional manifold without boundary, then

∫M dα = 0.


1.1.3 Cauchy’s formula

We propose to apply Stokes’ formula (1.5), using the following lemma.

Lemma 1.12 Let f : U → C be a holomorphicmap. Then the complex differ-ential form f dz is closed.

Proof We haved( f dz) = d( f dx+ i f dy) = ∂ f∂ydy∧ dx+ i ∂ f

∂x dx∧ dy. Thus∂ f∂y = i ∂ f

∂x implies thatd( f dz) = 0. �

By corollary 1.8, we thus also have the following.

Corollary 1.13 If f is holomorphic onU, the differential formfz−z0dz is closed

on U− {z0}.

Suppose now thatU contains a closed diskD. For everyz0 ∈ D, let Dε bethe open disk of radiusε centred atz0 which is contained inD for sufficientlysmall ε. ThenD − Dε is a manifold with boundary, whose boundary is theunion of the circle∂D and the circle of centrez0 and radiusε, the first with itsnatural orientation, the second with the opposite orientation. For holomorphicf , Stokes’ formula and corollary 1.1.3 then give the equality

1

2iπ

∫∂D

f (z)

z− z0dz= 1

2iπ

∫|z−z0|=ε

f (z)

z− z0dz. (1.6)

Furthermore, we have the following.

Lemma 1.14 If f is a function which is continuous at z0, then

limε→0

1

2iπ

∫|z−z0|=ε

f (z)

z− z0dz= f (z0).

Proof The circleof radiusε and centrez0 is parametrised by the mapγ : t �→z0+ εe2iπ t on the segment [0,1]. We haveγ ∗

(12iπ

f (z)z−z0dz

)= f (z0+ εe2iπ t )dt.

Thus,

1

2iπ

∫|z−z0|=ε

f (z)

z− z0dz=

∫ 1

0f (z0 + εe2iπ t )dt. (1.7)

But as f is continuous atz0, the functionsfε(t) = f (z0 + εe2iπ t ) converge


uniformly, asε tends to 0, to the constant function equal tof (z0). We thus have

limε→0

∫ 1

0f (z0 + εe2iπ t )dt =

∫ 1

0f (z0)dt = f (z0).

�Combining lemma 1.14 and equality (1.6), we now have

Theorem 1.15 (Cauchy’s formula) Let f be a holomorphic function on U andD a closed disk contained in U. Then for every point z0 in the interior of D,we have the equality

f (z0) = 1

2iπ

∫∂D

f (z)

z− z0dz. (1.8)

1.2 Holomorphic functions of several variables

1.2.1 Cauchy’s formula and analyticity

LetU be an open set ofCn, and let f : U → C be aC1map. Foru ∈ U , we havea canonical identificationTU,u

∼= Cn. We can thus generalise the notion of a

holomorphic function to higher dimensions.

Definition 1.16 The function f is said to be holomorphic if for every u∈ U,the differential

d fu ∈ Hom(TU,u, C) ∼= Hom(Cn, C)

isC-linear.

It is easy to prove that lemma 1.7 remains true in higher dimensions. Further-more, we have the three following characterisations of holomorphic functions.

Theorem 1.17The following three properties are equivalent for aC1 functionf :(i) f is holomorphic.(ii) In the neighbourhood of each point z0 ∈ U, f admits an expansion as apower series of the form

f (z0 + z) =∑I

αI zI , (1.9)

where I runs through the set of the n-tuples of integers(i1, . . . , i n) with ik ≥ 0,and zI := zi11 · · · zinn . The coefficients of the series (1.9) satisfy the following

1.2 Holomorphic functions of several variables 29

property: there exist R1 > 0, . . . , Rn > 0 such that the power series

∑I

|αI |r I

converges for every r1 < R1, . . . , rn < Rn.(iii) If D = {(ζ1, . . ., ζn)| |ζi − ai | ≤ αi } is a polydisk contained in U, then forevery z= (z1, . . ., zn) ∈ D0, we have the equality

f (z) =(

1

2iπ

)n ∫|ζi−ai |=αi

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn. (1.10)

In thepreceding formula, the integral is takenover aproduct of circles, equippedwith the orientation which is the product of the natural orientations.

Remark 1.18 Because of property (ii), holomorphic functions are also knownas complex analytic functions.

Proof The implications (iii)⇒(ii)⇒(i) are obvious: indeed, (iii)⇒(ii) is ob-tained, forζ in the product of circles{ζ | |ζi − ai| = αi } andz in the interior ofD, by expanding the functions

f (ζ )

(ζ1 − z1) · · · (ζn − zn)= f (ζ )

((ζ1 − a1) − (z1 − a1)) · · · ((ζn − an) − (zn − an))

as a power series inz1 − a1, . . . , zn − an whose coefficients are continuousfunctions ofζ . The uniform convergence inζ of this expansion then showsthat the integral (1.10) admits the corresponding power series expansion inz1 − a1, . . . , zn − an, so that (ii) holds for the function defined by (1.10).(ii)⇒(i) follows from the fact that every polynomial function ofz1, . . . , zn

is holomorphic, and that as the series (1.9) is a uniform limit of polynomialswhose derivatives also converge uniformly, its differential is the limit of thedifferentials of these polynomials. As the differential of each polynomial isC-linear, the same holds for the series, which is thus also holomorphic.It remains tosee that (i)⇒(iii),which isCauchy’s formula in several variables.

We can prove it by induction on the dimension, using Cauchy’s formula (1.8).We can also directly apply Stokes’ formula, using the following analogue oflemma 1.12.

Lemma1.19 If f is holomorphic, then thedifferential form f(z)dz1∧ · · · ∧dznis closed.


Theproductof circles∏

i {|ζi − zi|= ε} is contained inD for sufficiently smallε,and homotopic inD − ⋃

i {ζ | ζi = zi } to the product of circles∏

i {|ζi − ai | =αi },whichmeans that thereexistsanorientedcompactmanifoldM ofdimensionn and a differentiable map

φ : [0,1]× M → D − ∪i {ζ | ζi = zi }such thatφ|0×M is a diffeomorphism fromM to the first product of circles, andφ|1×M is a diffeomorphism fromM to the second product of circles, the firstisomorphism being compatible with the orientations, and the second changingthe orientation. We then deduce from lemma 1.19 that ifβ = f (ζ ) dζ1

ζ1−z1 ∧· · ·∧dζn

ζn−zn , the differential formφ∗β is closed on [0, 1]× M , and thus, by Stokes’formula, satisfies ∫

∂[0,1]×Mφ∗β = 0.

For ε sufficiently small, this gives the equality(

1

2iπ

)n ∫|ζi−ai |=αi

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn

=(

1

2iπ

)n ∫|ζi−zi |=ε

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn.

But the limit of the right-hand term asε tends to 0 is equal tof (z) by the sameargument as above.

�

Remark 1.20 The homotopy must have values in D− ⋃i {ζ | ζi = zi } and

not only in D, in order to guarantee that the formφ∗ f (ζ ) dζ1ζ1−z1 ∧ · · · ∧ dζn

ζn−zn isC1 in [0, 1]× M and to be able to apply Stokes’ formula.

1.2.2 Applications of Cauchy’s formula

Let us give some applications of theorem 1.17. To begin with, we have

Theorem 1.21 (The maximum principle ) Let f be a holomorphic function onan open subset U ofCn. If | f | admits a local maximum at a point u∈ U, thenf is constant in the neighbourhood of this point.

Proof Let R1, . . . , Rn be positive real numbers such that for everyεi ≤ Ri ,the polydiskDε· = {ζ ∈ C

n| |ζi − ui | ≤ εi } is contained inU . Then we have


Cauchy’s formula

f (u) =(

1

2iπ

)n ∫|ζi−ui |=εi

f (ζ )dζ1

ζ1 − u1∧ · · · ∧ dζn

ζn − un.

Parametrising the circles|ζi − ui | = εi by γi (t) = ui + εi e2iπ t , t ∈ [0,1], thiscan be written as

f (u) =∫ 1

0· · ·

∫ 1

0f (u1 + ε1e

2iπ t1, . . . ,un + εne2iπ tn)dt1 · · ·dtn. (1.11)

But we have the inequality∣∣∣∣∫ 1

0· · ·

∫ 1

0f (u1 + ε1e

2iπ t1, . . . ,un + εne2iπ tn)dt1 · · ·dtn

∣∣∣∣≤

∫ 1

0· · ·

∫ 1

0| f (u1 + ε1e

2iπ t1, . . . ,un + εne2iπ tn)| dt1 · · ·dtn, (1.12)

and equality holds if and only if the argument off (u1 + ε1e2iπ t1, . . . ,un +εne2iπ tn) is constant, necessarily equal to that off (u) by (1.11).Now, for sufficiently smallεi , we have by hypothesis

| f (u)| ≥ | f (u1 + ε1e2iπ t1, . . . ,un + εne

2iπ tn)| .

Combining this inequality with (1.11) and (1.12), we obtain

| f (u)| ≤∫ 1

0· · ·

∫ 1

0| f (u1 + ε1e

2iπ t1, . . .,un + εne2iπ tn)| dt1 · · ·dtn

≤∫ 1

0· · ·

∫ 1

0| f (u)| dt1 · · ·dtn = | f (u)| .

The equality of the two extreme terms then implies equality at every step; thefirst equality implies that the argument off is constant, equal to that off (u)on each product of circles as above, and the second equality shows that thefunction f must have constant modulus equal to| f (u) |, for sufficiently smallεi . Letting themultiradiusof thepolydisksDε· vary,wehave thusshown thatf isconstant, equal tof (u) on a neighbourhood ofupossiblyminus the hyperplanes{ζi = ui }, i.e. in fact constant in the neighbourhood ofu by continuity. �

Another essential application is the principle of analytic continuation.

Theorem 1.22Let U be a connected open set ofCn, and f a holomorphic

function on U. If f vanishes on an open set of U, then f is identically zero.


Proof This follows from the fact that by the characterisation (ii),f is in par-ticular analytic (i.e. locally equal to the sum of its Taylor series). We can thusapply the principle of analytic continuation tof . We recall that the latter isshown by noting that iff is analytic, the open set consisting of the points inwhose neighbourhoodf vanishes is equal to the closed set consisting of thepoints wheref and all its derivatives vanish. �

Let us now give some subtler applications of Cauchy’s formula (1.10) or itsgeneralisations. These theorems show that the possible singularities of a holo-morphic function cannot exist unless the function is not bounded (Riemann),and is not defined on the complement of an analytic subset of codimension 2(Hartogs).

Theorem 1.23 (Riemann) Let f be a holomorphic function defined onU − {z | z1 = 0}, where U is an open set ofCn. Then if f is locally boundedon U, f extends to a holomorphic map onU.

Proof Since this is a local statement, it suffices to show that ifU contains apolydiskD = {(z1, . . . , zn) ∈ C

n| |zi | ≤ ri } on which f is bounded, then wecan extendf to the points in the interior ofD. We propose to show that for apoint z in the interior ofD such thatz1 �= 0, Cauchy’s formula

f (z) =(

1

2iπ

)n ∫∂D

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn, (1.13)

where

∂D := {(ζ1, . . . , ζn)| |ζi | = ri , ∀i },holds. Note that the right-hand term in (1.13) is well-defined, since the integra-tion locus is contained in the locus of definition off .Letε1 ∈ R, 0< ε1 < |z1| be such that the closed disk of radiusε1 and centre

z1 is contained in the disk{ζ | |ζ | < r1}. Then the polydiskDε1 := {(ζ1, . . . , ζn)| |ζ1 − z1| ≤ ε1, |zi | ≤ ri , i ≥ 2}

is contained inD − {ζ1 = 0}, so that Cauchy’s formula gives

f (z) =(

1

2iπ

)n ∫∂Dε1

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn, (1.14)

where

∂Dε1 := {(ζ1, . . . , ζn)| |ζ1 − z1| = ε1, |ζi | = ri , i ≥ 2}.


Consider, also, the product of circles

∂D′ε := {(ζ1, . . . , ζn)| |ζ1| = ε, |ζi | = ri , i ≥ 2}.

Then whenε is sufficiently small,∂D − ∂Dε1 − ∂D′ε is the boundary of the

manifold

M = {(ζ1, . . . , ζn)| |ζ1 − z1| ≥ ε1, |ζ1| ≥ ε, |ζi | = ri , i ≥ 2},which is contained inD and intersects neither the hypersurface{ζ1 = 0} northe hypersurfaces{ζi = zi }. Here, the signs given to the components of theboundary are positive when the orientation as part of the boundary ofM co-incides with the natural orientation, negative otherwise. Stokes’ formula and(1.14) then give

f (z) =(

1

2iπ

)n[ ∫∂D

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn

−∫

∂D′ε

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn

].

The proof of formula (1.13) can then be finished using the following lemma.

Lemma 1.24When f is bounded, and for z such that z1 �= 0, |zi | < ri , wehave

limε→0

∫∂D′

ε

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn= 0. (1.15)

Proof Let us parametrise the product of circles∂D′ε by [0,1]

n, (t1, . . . , tn) �→(εe2iπ t1, r2e2iπ t2, . . . , rne2iπ tn). The integral (1.15) is thus equal to

(2iπ )n∫ 1

0· · ·

∫ 1

0εr2 · · · rn

∏je2iπ t j

f (εe2iπ t1, r2e2iπ t2, . . ., rne2iπ tn)

(εe2iπ t1 − z1) · · · (rne2iπ tn − zn)dt1· · ·dtn.

As f is bounded, under the hypotheses onz, the integrand in this formula tendsuniformly to 0 withε, and thus the integral in the formula tends to 0 withε.

�

As Cauchy’s formula (1.13) is proved, Riemann’s extension theorem followsimmediately, since it is clear that the function defined by the right-hand termin (1.13) extends holomorphically toD. �

To conclude this section, we will prove the following version of Hartogs’extension theorem.


Theorem 1.25Let U be an open set ofCn and f a holomorphic function onU − {z | z1 = z2 = 0}. Then f extends to a holomorphic function on U.

Remark 1.26 This implies the more general theorem mentioned above, usingtheorem 11.11, which proves that an analytic subset of codimension 2 can bestratified into smooth analytic submanifolds of codimension at least 2; thistheorem will be proved in section 11.1.1.

Proof Let D be a closed polydisk contained inU :

D = {(z1, . . . , zn)| |zi | ≤ ri }.Let z ∈ D − {ζ | ζ1 = ζ2 = 0}. As in the preceding proof, we will show thatCauchy’s formula

f (z) =(

1

2iπ

)n ∫∂D

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn(1.16)

is satisfied, and this will enable us to conclude, as above, that the functionf (z),given in the form of an integral as in (1.16), extends holomorphically toD. Letε1, ε2 be two positive real numbers, sufficiently small for the polydisk

Dε = {ζ | |ζi − zi | ≤ εi , i = 1, 2, |ζi | ≤ ri , i > 2}to be contained inD − {ζ | ζ1 = ζ2 = 0}. Then we have Cauchy’s formula

f (z) =(

1

2iπ

)n ∫∂Dε

f (ζ )dζ1

ζ1 − z1∧ · · · ∧ dζn

ζn − zn, (1.17)

where

∂Dε = {ζ | |ζi − zi | = εi , i = 1, 2, |ζi | = ri , i > 2}.It thus suffices to show that∂D − ∂Dε is a boundary in

D − ({ζ | ζ1 = ζ2 = 0} ∪⋃

i{ζ | ζi = zi }),

in order to apply Stokes’ formula and conclude that (1.16) holds.Let α1(t), α2(t), t ∈ [0,1], be two positive-valued differentiable functions

such thatαi (1)= εi , αi (0)= ri . For everyt ∈ [0,1], let

∂Dt = {ζ | |ζi − tzi | = αi (t), i = 1, 2, |ζi | = ri , i > 2}.

Lemma 1.27 For a suitable choice of functionsα1, α2, ∂Dt is contained in

D − ({ζ | ζ1 = ζ2 = 0} ∪⋃

i{ζ | ζi = zi })

for every t∈ [0,1].

1.3 The equation∂g∂z = f 35

Proof Firstly, ∂Dt still lies in D if αi (t) + t |zi | ≤ ri , i = 1, 2. Moreover,∂Dt still lies in D − ⋃

i {ζ | ζi = zi } if αi (t) �= (1− t) |zi |, i = 1, 2. Now,note that (1− t) |zi | < ri − t |zi |, since|zi | < ri . Furthermore, the conditionsαi (t) ≤ ri − t |zi | andαi (t) > (1− t) |zi | are both satisfied fort = 1 andt = 0.It thus suffices to take functionsαi (t) satisfying

(1− t) |zi | < αi (t) ≤ ri − t |zi |, αi (0)= ri , αi (1)= εi .

It remains to see thatDt does not meet{ζ | ζ1 = ζ2 = 0} for any t ∈ [0,1],for a suitable choice of the pair (α1, α2). But Dt meets{ζ | ζ1 = ζ2 = 0} if wehaveαi (t) = t |zi | for i = 1 and 2. For fixedt , this imposes two conditions onthe pair (α1, α2), andt varies in a segment, so it is clear that this last conditionis not satisfied by a pair of sufficiently general functions. �

Lemma 1.27 gives a differentiable homotopy inD − ({ζ | ζ1 = ζ2 = 0} ∪{ζ | ζi = zi }) from ∂D to ∂Dε , so we can conclude by Stokes’ formula that(1.16) holds. Thus theorem 1.25 is proved. �

1.3 The equation∂g∂z = f

The following theorem will play an essential role in the proof of the localexactness of the operator∂.

Theorem 1.28Let f be aCk function (for k≥ 1) on an open set ofC. Then,locally on this open set, there exists aCk function g (for k≥ 1), such that

∂g

∂z= f. (1.18)

Remark 1.29 Such a function g is defined up to the addition of a holomorphicfunction.

Proof As the statement is local, we may assume thatf has compact support,and thus is defined andCk onC. Now set

g = 1

2iπ

∫C

f (ζ )

ζ − zdζ ∧ dζ .

�

Remark 1.30 This is a singular integral. By definition, it is equal to the limit,asε tends to0, of the integrals

1

2iπ

∫C−Dε

f (ζ )

ζ − zdζ ∧ dζ ,


where Dε is a disk of radiusε centred at z. It is easy to see that this limit exists(the function 1

ζ−z is L1).

Making the change of variableζ ′ = ζ − z, we also have

g(z) = limε→0

gε(z), gε(z) = 1

2iπ

∫Cε

f (ζ ′ + z)

ζ ′ dζ ′ ∧ dζ ′,

where

Cε = C − D′ε,

andD′ε is a disk of radiusε centred at 0. The convergence of thegε whenε

tends to 0 is uniform inz. Moreover, we can differentiateunder the integralsign the (non-singular) integral defininggε

∂gε

∂z= 1

2iπ

∫Cε

∂ f (ζ ′ + z)

∂z

dζ ′ ∧ dζ ′

ζ ′ .

As ∂ f (ζ ′+z)∂z is Ck−1, with k− 1 ≥ 0, the functions∂gε

∂z convergeuniformly, andwe conclude thatg is at leastC1 and satisfies

∂g

∂z= 1

2iπ

∫C

∂ f (ζ ′ + z)

∂z

dζ ′ ∧ dζ ′

ζ ′ .

By induction onk, the same argument actually shows thatg is Ck. Thus, itremains to show the equality∂g

∂z = f . Again making the change of variableζ = ζ ′ + z, we have

∂g

∂z(z) = lim

ε→0

1

2iπ

∫C−Dε

∂ f

∂ζ(ζ )

dζ ∧ dζ

ζ − z. (1.19)

Now, we have the equality onC − Dε

∂ f

∂ζ(ζ )

dζ ∧ dζ

ζ − z= −d

(fdζ

ζ − z

);

indeed, for a differentiable functionφ(ζ ) we know thatdφ = ∂φ

∂ζdζ + ∂φ

∂ζdζ ,

and thus

d(φdζ ) = −∂φ

∂ζdζ ∧ dζ .

Stokes’ formula thus gives

1

2iπ

∫C−Dε

∂ f

∂ζ(ζ )

dζ ∧ dζ

ζ − z= 1

2iπ

∫∂Dε

f (ζ )dζ

ζ − z. (1.20)

Exercises 37

Using lemma 1.14 and the equalities (1.19), (1.20) we have thus proved theequality (1.18).

Exercises

1. Letφ : U → V be a holomorphic map from an open subset ofCn to an open

subset ofCn. Show that the set

R= {x ∈ U | dφx is not an isomorphism}is defined inU by exactly one holomorphic equation.This set is called the ramification divisor ofφ, when it is different fromU .

2. Let f be a holomorphic function defined over an open subsetU of Cn. We

assume thatf does not vanish outside the set

{z= (z1, . . . , zn) ∈ U | z1 = z2 = 0}.Show thatf does not vanish at any point ofU .

3. Let f be a meromorphic function defined on an open subsetU of C. Thismeans thatf is locally the quotient of two holomorphic functions.(a) Show that for any compact subsetK ⊂ U , the number of zeros or polesof f in K is finite.(b) Let x ∈ U . Show that there exists an integerkx ∈ Z such thatf can bewritten as (z − x)kxφ in a neighbourhood ofx, with φ holomorphic andinvertible (that is non-zero).The divisor of f is defined as the locally finite sum∑

x∈Ukxx.

(c) Letx ∈ U andD ⊂ U be a disk centred inx, such thatx is the only poleor zero of f in D. Show that

kx =∫

∂D

1

2iπ

d f

f.

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