CONTEMPORARY MATHEMATICS

207

Birational Algebraic Geometry

A Conference on Algebraic Geometry in Memory of

Wei-Liang Chow (1911-1995) April 11-14, 1996

Japan-U.S. Mathematics Institute Johns Hopkins University

Yujiro Kawamoto Vyacheslav V. Shokurov

Editors

Selected Titles in This Series

207 Yujiro Kawamata and Vyacheslav V. Shokurov, Editors, Birational algebraic geometry: A conference on algebraic geometry in memory of Wei-Liang Chow (1911-1995), 1997

206 Adam Koranyi, Editor, Harmonic functions on trees and buildings, 1997 205 Paulo D. Cordaro and Howard Jacobowitz, Editors, Multidimensional complex

analysis and partial differential equations: A collection of papers in honor of Frant;ois Treves, 1997

204 Yair Censor and Simeon Reich, Editors, Recent developments in optimization theory and nonlinear analysis, 1997

203 Hanna Nencka and Jean-Pierre Bourguignon, Editors, Geometry and nature: In memory of W. K. Clifford, 1997

202 Jean-Louis Loday, James D. Stasheff, and Alexander A. Voronov, Editors, Operads: Proceedings of Renaissance Conferences, 1997

201 J. R. Quine and Peter Sarnak, Editors, Extremal Riemann surfaces, 1997 200 F. Dias, J.-M. Ghidaglia, and J.-C. Saut, Editors, Mathematical problems in the

theory of water waves, 1996 199 G. Banaszak, W. Gajda, and P. Krason, Editors, Algebraic K-theory, 1996 198 Donald G. Saari and Zhihong Xia, Editors, Hamiltonian dynamics and celestial

mechanics, 1996 197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry,

1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective

endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, Lie algebras and their

representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and

related topics, 1996 192 Tomek Bartoszyriski and Marion Scheepers, Editors, Set theory, 1995 191 Tuong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally,

Jr., Editors, Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F.

Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A.M. Marcantognini, G. A. Mendoza, M.D. Moran, A. Octavia, and W. 0.

Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors,

Homotopy theory and its applications, 1995 187 G. W. Brumftel and H. M. Hilden, S£(2) representations of finitely presented groups,

1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka Ihara,

and Helmut Voelklein, Editors, Recent developments in the inverse Galois problem, 1995

185 Raul E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberto Salinas, Editors, Multivariable operator theory, 1995

184 L.A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International Conference on Algebra, 1995

183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 1995

182 Selman Akbulut, Editor, Real algebraic geometry and topology, 1995 181 Mila Cenkl and Haynes Miller, Editors, The Cech Centennial, 1995 180 David E. Keyes and Jinchao Xu, Editors, Domain decomposition methods in

scientific and engineering computing, 1994 (Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/207

Birational Algebraic Geometry

In Memoriam: Wei-Liang Chow

1911-1995

Photograph courtesy of the Chow family.

CoNTEMPORARY MATHEMATICS

207

Birational Algebraic Geometry

A Conference on Algebraic Geometry in Memory of

Wei-Liang Chow (1911-1995) April 11-14, 1996

Japan-U.S. Mathematics Institute Johns Hopkins University

Yujiro Kawamoto Vyacheslav V. Shokurov

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeThrck, managing editor

Andy Magid Michael Vogelius Clark Robinson Peter M. Winkler

This volume features proceedings from the JAMI Conference on Birational Algebraic Geometry in Memory of Wei-Liang Chow (1911-1995) held at the Johns Hopkins Univer-sity in Baltimore, MD, on April 11-14, 1996. Many of the lectures given at the meeting are represented here in addition to some work that was invited but not presented at the meeting.

Support was provided by Johns Hopkins University, the Japan-U.S. Mathematical In-stitute (JAMI), the Japan Society for the Promotion of Science (JSPS Grant MPCR-315), the National Science Foundation (NSF Grant INT-9416927), and the National Security Agency (NSA Grant MDA-9049511075). The grants of JSPS and NSF were under the United States-Japan Cooperative Science Program.

1991 Mathematics Subject Classification. Primary 14-Q6, 14E15, 14E30, 14J40, 14J45, 14J60, 32E10.

Library of Congress Cataloging-in-Publication Data Birational algebraic geometry : a conference on algebraic geometry in memory of Wei-Liang Chow {1911-1995), April 11-14, 1996, Japan-U.S. Mathematics Institute, Johns Hopkins University / Yujiro Kawamata, Vyacheslav V. Shokurov, editors.

p. em. Includes bibliographical references. ISBN 0-8218-0769-2 {alk. paper) 1. Geometry, Algebraic-Congresses. I. Chow, Wei-Liang, 1911-1995. II. Kawamata, Yujiro,

1952- . III. Shokurov, Vyacheslav V., 1950- . QA564.B49 1997

516.3'5--dc21 97-7968 CIP

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. R<equests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. R<equests can also be made by e-mail to reprint-permissionOams.org.

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§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

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Contents

Preface

Remarks on Chow JUN-ICHI IGUSA AND SERGE LANG

Chow's Bibliography

On the second adjunction mapping and the very ampleness of the triadjoint bundle

ix

xi

xix

MAURO C. BELTRAMETTI AND ANDREW J. SOMMESE 1

Stein small deformations of strictly pseudoconvex surfaces FEDOR A. BOGOMOLOV AND BRUNO DE OLIVEIRA 25

On the rationality of non-Gorenstein Q-Fano 3-folds with an integer Fano index IVAN CHELTSOV 43

Nilpotent cones and sheaves on K3 surfaces RON DONAGI, LAWRENCE EIN, AND ROBERT LAZARSFELD 51

Minimal, canonical and log-canonical models of hypersurface singularities SHIHOKO ISHII 63

Subadjunction of log canonical divisors for a subvariety of codimension 2 YUJIRO KAWAMATA 79

Relative De Rham complex for non-smooth morphisms SANDOR J. Kov A.cs 89

A note on moderate abelian fibrations KEIJI 0GUISO 101

On extremal contractions from threefolds to surfaces: the case of one non-Gorenstein point YURI G. PROKHOROV

Letters of a birationalist. I. A projectivity criterion V. V. SHOKUROV

vii

119.

143

Preface

The papers in these Proceedings grew out of lectures given at the JAMI Confer-ence on Birational Algebraic Geometry in Memory of Wei-Liang Chow (1911-1995) held at the Johns Hopkins University in Baltimore, Maryland on April11-14, 1996. Many of the lectures given at the meeting are represented here; in addition, some work that was invited but not presented at the meeting appears. We also think that it is relevant to include an address of Jun-ichi Igusa and some remarks about Wei-Liang Chow by him and Serge Lang. The meeting as well as these Proceedings brought into evidence many of the directions in which the study of birational alge-braic geometry is headed: problems on special models as Fanos and their fibrations, adjunctions and subadjunction formuli, projectivity and projective embeddings.

Note that the paper on "Nilpotent cones and sheaves on K3 surfaces" by R. Donagi, L. Ein and R. Lazarsfeld" does not really prove any substantive theorems. However, it points out an amusing connection between the Hitchin dynamical sys-tem on the cotangent bundle of the moduli spaces of bundles on a curve and in-stances of the Mukai moduli spaces of simple sheaves on a K3. In the Shokurov's note on "A projectivity criterion," main conjectures are derived from the LMMP (Log Minimal Model Program) so they are established for the dimensions~ 3, and in the dimensions ~ 4 results depend on a progress in the LMMP.

The conference took place toward the end of the special year of the Japan-U.S. Mathematical Institute (JAMI) of the Johns Hopkins University entitled "Clas-sification of Algebraic Varieties". Thanks to the Johns Hopkins University, its Japan-U.S. Mathematical Institute (JAMI), the Japan Society for the Promotion of Science (JSPS Grant MPCR-315), the National Science Foundation (NSF Grant INT-9416927) and the National Security Agency (NSA Grant MDA-9049511075) for their generous support. The grants of JSPS and NSF are under the United States-Japan Cooperative Science Program.

Yujiro Kawamata and Vyacheslav V. Shokurov

ix

REMARKS ON CHOW

From an opening speech at the conference on Birational Algebraic Geometry in memory of Wei-Liang Chow

by Jun-ichi lgusa

As everyone knows, Chow was a great mathematician and a highly esteemed professor at Johns Hopkins University since the late 1940's, but Wei-Liang Chow was also a great man who overcame enormous difficulties both personal and pro-fessional.

We go back to 1936 when Professor Chow was in Germany as a student. He proved the main theorem on Chow forms in the spring, then married in July. A few weeks later Chow went back to China with Mrs. Chow and started teaching at a university in Nanjing. But only a year after in 1937 Imperial Japan started an invasion of China. Professor Chow and his family suffered 8 years under the occupation of the army of Imperial Japan and two more years during the civil war in China before they came to the United States in 1947. From age 25 to 35 it was, therefore, almost impossible for him to do any mathematics. He told me on several occasions how his family spent those difficult years in Shanghai. I would like to repeat that Professor Chow lost 10 years in his career. In fact these were the critical 10 years from age 25 to 35 when he could have been the most productive. One should also not overlook the fact that he later gave Hopkins 10 years of his precious time to serve as chairman. I hope that you will now appreciate the greatness of Wei-Liang Chow as a man as well as his greatness as a mathematician.

Table Speech on Professor Wei-Liang Chow

by Jun-ichi lgusa

I would like to thank the organizers very much for allocating this time for me to say a few words about Professor Chow. Since I have already written an article about Professor Chow (see Lang's comments below), I shall talk about something not mentioned in that article.

The first time I met Professor Chow was in early April of 1954 during a confer-ence at Princeton in honor of Professor Lefschetz. However a rather extraordinary thing happened about 2 months earlier. At that time I was at Harvard with Profes-sor Zariski. I might mention that Abhyankar was also there as a student of Professor Zariski. Zariski asked me, during one of our mathematical conversations, if I could prove the arithmetic normality of the Grassmann variety, i.e., the fact that the ring generated by Pliicker coordinates is integrally closed. This is equivalent to what Hermann Weyl called the first main theorem on vector invariants for SL(n). and it implies the fact that every Chow form can be expressed as a polynomial in the Pliicker coordinates of the variable vectors involved. It was a classical theorem in characteristic 0, but was at that time an unsolved problem in characteristic p > 0. I thought about that problem for a while and produced a proof. I presented my proof at Zariski's seminar sometime in January of that year, i.e., in 1954. One or two weeks later, ProfessorS, who was visiting Harvard that year, went to Prince-ton to negotiate a position as a visiting professor. After coming back, he said, 'I

xi

xii REMARKS ON CHOW

saw Chow at Princeton and told him that you proved the arithmetic normality of the Grassmann variety. But, you know, Chow completely denied that you proved it." That was an extremely strong statement. I might mention that I was not an unknown mathematician to Professor Chow at that time. In fact, two years before that, he and Professor Andre Weil wrote separate papers which both quoted my dissertation on Picard varieties in their first sentences. Furthermore, my proof was already discussed at Zariski's seminar. So I was puzzled and asked Professor S. why Chow completely denied my proof. To this he said, "Chow said that he once thought that he proved the arithmetic normality but found that he overlooked a subtle point. Therefore if there is no noticeable new idea in lgusa's proof, then lgusa must have made the same or a similar mistake." That was the explanation of his strong statement. Needless to say, I had profound respect for Professor Chow in view of the achievements such as his brilliant work on Chow forms. Therefore, after hearing his comment, I carefully re-examined my proof. You can now guess the outcome. I found that Professor Chow was absolutely correct. My proof was not a proof at all. It completely collapsed. In view of fairness to Professors Zariski, Abhyankar, and the others who attended the seminar, I should add that my English at that time, and to some extent even now, was terrible; that might have been the reason why they overlooked my error. At any rate, I worked hard about a week or so in February of that year and fortunately found a correct proof. It was based on the effective use of what are now called standard monomials and a lemma by Hodge on such monomials. In the end, Professor Chow saved me from writing an unworthy paper containing a completely erroneous proof. I have chosen this story because it clearly tells one aspect of Professor Chow, i.e., how sharp and also how confident he was.

As I have said, I met Professor Chow for the first time in early April of 1954. A year and a half later I came to Hopkins to work with Professor Chow, and since then I have been closely associated with him for 40 years. This was not entirely accidental. I have stayed at Hopkins mainly because of Professor Chow. I would like to mention that his name "Wei" is a rather strange Chinese character - strange to a Japanese because one does not use it in Japan. It is a combination of greatness and "fire." Professor Chow was indeed a sparkling great man and also a great man with unusual warmth. That might have been the reason why many people were attracted to him. I feel truly fortunate to have known Professor Chow from the time when he was in his early 40's until he passed away last summer. I would like to say further that we, i.e., my wife Yoshie and I, are extremely grateful for the warm treatment by Professor Chow and his wife Margot during those memorable years.

Thank you very much!

From a letter by Serge Lang

Having written fairly extensive comments on Chow's (see below) work made me think back to that wonderful period of the fifties when I was· just entering mathematics and had the luck to know Chow at his most creative. It was very exciting to learn of this work as it was progressing, and I used some of it in an essential way. Chow was among the finest persons I ever knew, with impeccable

REMARKS ON CHOW xiii

judgment, honesty and kindness. For a young inexperienced mathematician as I was, it was really a great boost and pleasure for me to deal with Chow.

It is a great pity that, as Igusa has related in his comments (Chow, p. 1124], lack of money prevented his university1 and department from keeping intact the powerful group of mathematicians he had attracted at Hopkins. But Chow's contributions to research, to his department, and to people on the human level, will remain strong in our memories and will be recognized by those who follow us.

Comments on Chow's Works 2

Serge Lang Van der Waerden's pre-war series of articles began an algebraization of Italian

algebraic geometry. I was born into algebraic geometry in the immediate post war period. This period was mostly characterized by the work of Chevalley, Chow, Weil (starting with his Foundations and his books on correspondences and abelian varieties), and Zariski. In the fifties, there was a constant exchange of manuscripts among the main contributors of that period. I shall describe briefly some of Chow's contributions. I'll comment here mostly on some of Chow's works in algebraic geometry, which I know best.

§ 1. Chow coordinates One of Chow's most influential works was also his first, namely the construction

of the Chow form, in a paper written jointly with van der Waerden (ChW 37a] (see Chow's papers and books in Chow's Bibliography), but it was explicitly mentioned that the material dealing with Chow forms was due to Chow. To each projective variety, Chow saw how to associate a homogeneous polynomial in such a way that the association extends to a homomorphism from the additive monoid of effective cycles in projective space to the multiplicative monoid of homogeneous polynomials, and the association is compatible with the Zariski topology. In other words, if one cycle is a specialization of another, then the associated Chow form is also a specialization. Thus varieties of given degree in a given projective space decompose into a finite number of algebraic families, called Chow families. The coefficients of the Chow form are called the Chow coordinates of the cycle, or of the variety. Two decades later, he noted that the Chow coordinates can be used to generate the smallest field of definition of a divisor (Ch 50a]. He also applied the Chow form to a study of algebraic families when he gives a criterion for local analytic equivalence (Ch 50b]. He was to use them all his life, in various contexts dealing with algebraic families.

In Grothendieck's development of algebraic geometry, Chow coordinates were bypassed by Grothendieck's construction of Hilbert schemes, whereby two schemes are in the same family whenever they have the same Hilbert polynomial. The Hilbert schemes can be used more advantageously than the Chow families in some cases. However, as frequently happens in mathematics, neither is a substitute for the other in all cases. In recent times, say during the last decade, Chow forms and coordinates have made a reappearance due to a renewed emphasis on explicit constructions needed to make theorems effective (rather than having non-effective

1The Johns Hopkins University 2 First published in Notices of the AMS, vol. 43 (1996), No. 10, pp. 1117-1124 published by

the American Mathematical Society.

xiv REMARKS ON CHOW

existence proofs, say), and for computational aspects of algebraic geometry whereby one wants not only theoretical effectiveness but good bounds for solutions of alge-braic geometric problems as functions of bounds on the data. Projective construc-tions such as Chow's are very well suited for such purposes. Thus Chow coordinates reappeared both in general algebraic geometry, and also in Arakelov theory and in diophantine applications. The Chow coordinates can be used for example to define the height of a variety, and to compare it to other heights constructed by more intrinsic, non-projective methods as in [Ph 91], [Ph 94], [Ph 95]. They were used further in Arakelov theory by Bin Wang [Wa 96].

Chow coordinates were also used to prove a conjecture of Lie on a converse to Abel's theorem. See the papers by Wirtinger [Wi 38] and Chern [Che 83].

§2. Abelian varieties and group varieties (a) Projective construction of the Jacobian variety. In the fifties, Chow

contributed in a major way to the general algebraic theory of abelian varieties due to Weil (who algebraicized the transcendental arguments of the Italian school, especially Castelnuovo). For one thing, Chow gave a construction of the Jacobian variety by projective methods, giving the projective embedding directly and also effectively [Ch 54]. The construction also shows that when a curve moves in an algebraic family, then the Jacobian also moves along in a corresponding family.

(b) The Picard variety. Chow complemented Igusa's transcendental con-struction of the Picard variety by showing how this variety behaves well in alge-braic systems, using his "associated form" [Ch 52b]. He announced an algebraic construction of the Picard variety in a "forthcoming paper." Indeed, such a paper circulated as an unpublished manuscript a few years later [Ch 55c], but was never published as far as I know.

(c) Fixed part of an algebraic system. Chow also developed a theory of algebraic systems of abelian varieties, defining the fixed part of such systems, i.e. that part which does not depend genuinely on the parameters [Ch 55a,b]. His notion of fixed part was used by others in an essential way, e.g. by Lang-Neron, who proved that for an abelian variety A defined over a function field K, the group of rational points of A in K modulo the group of points of the fixed part is finitely generated [LaN 59]. This is a relative version of the Mordell-Weil theorem.

(d) Field of definition. Chow gave conditions under which an abelian variety defined over an extension of a field k can actually be defined over k itself [Ch 55a,b]. Chow's idea was extended by Lang [La 55] to give such a criterion for all varieties, not just abelian varieties, and Weil reformulated the criterion in terms of cohomology (splitting a cocyde) [We 56].

§3. Homogeneous spaces (a) Projective embedding of homogeneous spaces. Chow extended the

Lefschetz-Weil proof of the projective embedding of abelian varieties to the case of homogeneous spaces over arbitrary group varieties, which may not be complete [Ch 57 a]. Chow's proof has been overlooked in recent years, even though interest in projective constructions has been reawakened, but I expect Chow's proof to make it back to the front burner soon, just like his other contributions.

(b) Algebraic properties. Chow's paper [Ch 49b] dealt with the geometry of homogeneous spaces. The main aim of this paper is to characterize the group by geometric properties. The latter could refer to the lines in a space, as in projective geometry, or to certain kinds of matrices, such as symmetric matrices. For instance,

REMARKS ON CHOW XV

a typical theorem says: Any bijective adjacence preserving transformation of the space of a polar system with itself is due to a transformation of the basic group, provided that the order of the space is greater than 1. Birational geometry is considered in this context.

§4. The Chow ring In topology, intersection theory holds for the homology ring. In 1956, Chow

defined rational equivalence between cycles on an algebraic variety, he defined the intersection product for such classes, and thus obtained the Chow ring [Ch 56a], which proved to be just as fundamental in algebraic geometry as its topological counterpart. The latest surfacing of the Chow groups is in [GoM 96].

§5. Algebraic geometry over rings In the late fifties began the extension of algebraic geometry over fields to alge-

braic geometry over rings of various type, partly to deal with algebraic or analytic families, but partly because of the motivation from number theory, where one deals with local Dedekind rings, p-adic rings, and more generally complete Noetherian local rings. Chow contributed to this extension in several ways. Of course, in the sixties Grothendieck vastly and systematically went much further in this direction, but it is often forgotten that the process had begun earlier. I shall mention here some of Chow's contributions in this direction.

(a) Connectedness theorem. In 1951 Zariski had proved a general connect-edness theorem for specializations of connected algebraic sets. Zariski based his proof on an algebraic theory of holomorphic functions which he developed for this purpose. In [Ch 57b] and [Ch 59], Chow gave a proof of a generalization over ar-bitrary complete Noetherian local domains, based on much simpler techniques of algebraic geometry, especially the Chow form.

(b) Uniqueness of the integral model of a curve. The paper [ChL 57c] proved the uniqueness of the model of a curve of genus ~ 1 and an abelian variety over a discrete valuation ring, in the case of non-degenerate reduction.

(c) Cohomology. Invoking the theory of deformations of complex analytic structures by Kodaira-Spencer, the connectedness theorem, and Igusa's work on moduli spaces of elliptic curves, Chow and Igusa proved the upper semicontinuity of the cohomology over a broad class of Noetherian local domains [Chi 58c]. Semi-continuity was proved subsequently in the complex analytic case by Grauert, and by Grothendieck in more general algebraic settings. However, Chow's and Igusa's contribution did not get the credit they deserved. Cf. [Ha 77], Chapter 3, §12, and the bibliographical references given there, referring to work in the sixties, but not to Chow-Igusa.

(d) Bertini's theorem. During that same period in the late fifties, Chow extended Bertini's theorem to local domains [Ch 58b].

(e) Unmixedness theorem. A homogeneous ideal defining a projective va-riety is said to be unmixed if it has no embedded prime divisors. Chow proved that the Segre product of two unmixed ideals is also unmixed, under fairly general conditions, in a ring setting [Ch 64].

§6. Algebraicity of analytic objects Chow was concerned over many years with the algebraicity of certain complex

analytic objects. We mention two important instances. (a) Meromorphic mappings and formal functions. In 1949, Chow p~oved

the fundamental fact, very frequently used from then on, that a complex analytic

xvi REMARKS ON CHOW

subvariety of projective space is actually algebraic [Ch 49a]. Twenty years later, he came back to similar questions, and proved in the context of homogeneous varieties that a meromorphic map is algebraic [Ch 69]. Remarkably, and wonderfully, almost twenty years after that, he came back once more to the subject and completed it in an important point [Ch 86]. I quote from the introduction to this paper, which shows how Chow was still lively mathematically: "Let X be a homogeneous algebraic variety on which a group G acts, and let Z be a subvariety of positive dimension. Assume that Z generates X [in a sense which Chow makes precise] ... One asks whether a formal rational function on X along Z is the restriction along Z of an algebraic function (or even a rational function) on X. In a paper [Ch 69] some years ago, the author gave an affirmative answer to this question, under the assumption that the subvariety Z is complete, but only for the complex-analytic case with the formal function replaced by the usual analytic function defined in a neighborhood of Z. The question remains whether the result holds also for the formal functions in the abstract case over any ground field. We had then some thoughts on this question, but we did not pursue them any further as we did not see a way to reach the desired conclusion at the time. In a recent paper [3], Faltings raised this same question and gave a partial answer to it in a slightly different formulation. This result of Faltings led us to reconsider this question again, and this time we are more fortunate. In fact, we have been able not only to solve the problem, but also to do it by using essentially the same method we used in our original paper."

(b) Analytic surfaces. In a paper with Kodaira, it was proved that a Kahler surface with two algebraically independent meromorphic functions is a non-singular algebraic surface [ChK 52c].

§7. Other works in algebraic geometry Chow's papers in algebraic geometry include a number of others, which, as I

already asserted, I am less well acquainted with, and won't comment upon, such as his paper on the braid group [Ch 48], on the fundamental group of a variety [Ch 52d], on rational dissections [Ch 56b], and on real traces of varieties [Ch 63].

§8. PDE Chow's very early paper on systems of linear partial differential equations of

first order [Ch 39c] gives a generalization of a theorem of Caratheodory on the foun-dations of thermodynamics. This paper had effects not well known to the present generation of mathematicians, including me. It was only just now brought to my attention. An anonymous colleague wrote to the editor of the present collection of articles on Chow's work: "This paper essentially asserts the identity of the integral submanifold of a set of vector fields and the integral submanifold of the Lie algebra generated by the set of vector fields. This is widely known as "Chow's theorem" in nonlinear control theory, and is the basis for the study of the controllability problem in nonlinear systems. Controllability refers to the existence of an input signal that drives the state of a system from a given initial state to a desired terminal state. A more detailed exposition of the role of Chow's theorem, with several references, is provided in the survey paper [Br 76]."

REMARKS ON CHOW xvii

References

[Br 76] R. W. BROCKETT, Nonlinear systems and differential geometry, Proc. IEEE 64 (1976) pp. 61-71

[Che 83] S. S. CHERN, Web Geometry, AMS Proc. Symp. in Pure Math. Vol. 39 (1983), pp. 3-10

[GoM 96] B.B. GORDON and J.P. MURRE, Chow Motives of elliptic Modular Surfaces and Threefolds, preprint, Mathematics Institute, University of Leiden, 1996

[Ha 77] R. HARTSHORNE, Algebraic Geometry, Springer Verlag, 1977 [La 55] S. LANG, Abelian Varieties over finite fields, Proc. Nat. Acad. Sci. USA 41 No. 3 (1955)

pp. 174-176 [LaN 59]

[Ph 91] [Ph 94]

[Ph 95]

[Wa 96] [We 56] [Chow]

[Wi38]

S. LANG and A. NERON, Rational points of abelian varieties over function fields, Amer. J. Math. 81 No. 1 (1959) pp. 95-118 P. PHILIPPON, Sur des hauteurs alternatives I, Math. Ann. 289 (1991) pp. 255-283 P. PHILIPPON, Sur des hauteurs alternatives II, Ann. Institut Fourier 44 No.4 (1994) pp. 1043-1065 P. PHILIPPON, Sur des hauteurs alternatives III, J. Math. Pures Appl. 74 (1995) pp. 345-365 B. WANG, Mazur's incidence structure for projective space, preprint, 1996 A. WElL, The field of definition of a variety, Amer. J. Math. 78 No.3 (1956) pp. 509-524 W.S. WILSON, S.S. CHERN, S. ABHYANKAR, S. LANG, J.-1. IGUSA, Wei-Liang Chow, in Notices ofthe AMS, vol. 43 (1996), No. 10, pp. 1117-1124 W. WIRTINGER, Lies Translationsmannigfaltigkeiten une Abelsche Integrale, Monat. Math. u. Physik, 46 (1938) pp. 384-443

Chow's Bibliography

References

[ChW 37a] (with van der Waerden) Zur algebraische Geometrie IX, Math. Ann. 113 (1937) pp. 692-704

[Ch 37b]

[Ch 39a]

[Ch 39b]

[Ch 39c]

[Ch 40] [Ch 48] [Ch 49a] [Ch 49b]

[Ch 49c]

[Ch 49d]

Die geometrische Theorie der algebraischen Funktionen fiir beliebige vollkommene Korper, Math. Ann. 114 (1937) pp. 655-682 Einfacher topologischer Beweis des Fundamentalsatzers der Algebra, Math. Ann. 116 (1939) p. 463 Ueber die Multiplizitiit der Schnittpunkte von Hyperfliichen, Math. Ann. 116 (1939) pp. 598-601 Uber systemen von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939) pp. 98-108 On Electric Networks, J. Chinese Math. Soc. 2 (1940) pp. 321-339 On the algebraical braid group, Ann. Math. 49 No. 3 (1948) pp. 654-658 On compact complex analytic varieties, Amer. J. Math. 71 No.4 (1949) pp. 893-914 On the geometry of algebraic homogeneous spaces, Ann. Math. 50 No. 1 (1949) pp. 32-67 Ueber die Losbarkeit gewisser algebraischen Gleichungssysteme, Comm. Math. Helv. 23 (1949) pp. 76-79 On the genus of curves of an algebraic system, Trans. Amer. Math. Soc., 65 (1949) pp. 137-140

[Ch 50a] On the defining field of a divisor in an algebraic variety, Proc. Amer. Math. Soc. 1 No. 6 (1950) pp. 797-799

[Ch 50b] Algebraic systems of positive cycles in an algebraic variety, Amer. J. Math. 72 No. 2 (1950) pp. 247-283

[Ch 52a] On the quotient variety of an Abelian variety, Proc. Nat. Acad. Sci. (USA) 38 (1952) pp. 1039-1044

[Ch 52b] On Picard varieties, Amer. J. Math. 74 No.4 (1952) pp. 895-909 [ChK 52c] (with Kodaira) On analytic surfaces with two independent meromorphic functions,

[Ch 52d] [Ch 52e] [Ch 54] [Ch 55a] [Ch 55b] [Ch 55c] [Ch 56a]

Proc. NAS 38 No. 4 (1952) pp. 319-325 On the fundamental group of an algebraic variety, Amer. J. Math. 74 (1952) pp. 726-736 On the quotient variety of an Abelian variety, Proc. NAS 38 (1952) pp. 1039-1044 The Jacobian variety of an algebraic curve, Amer. J. Math. 76 No.2 (1954) pp. 453-476 On Abelian varieties over function fields, Proc. NAS 41 (1955) pp. 582-586 Abelian varieties over function fields, Trans. Amer. Math. Soc. 78 (1955) pp. 253-275 Abstract theory of the Picard and Albanese varieties, unpublished manuscript On equivalence classes of cycles in an algebraic variety, Ann. Math. 64 No. 3 (1956) pp. 450-479

[Ch 56b] Algebraic varieties with rational dissections, Proc. NAS 42 (1956) pp. 116-119 [Ch 57 a] On the projective embedding of homogeneous spaces, Lefschetz conference volume, Al-

gebraic Geometry and Topology, Princeton University Press (1957) pp. 122-128 [Ch 57b] On the principle of degeneration in algebraic geometry, Ann. Math. 66 (1957) pp. 70-79 [ChL 57c] (With S. Lang) On the birational equivalence of curves under specialization, Amer. J.

Math. 79 (1957) pp. 649-652 [Ch 58a] The criterion for unit multiplicity and a generalization of Hensel's lemma, Amer. J.

Math. 80 (1958) pp. 539-552 [Ch 58b] On the theorem of Bertini for local domains, Proc. NAS 44 No. 6 (1958) pp. 580-584 [Chi 58c] (with Igusa) Cohomology theory of varieties over rings, Proc. NAS 44 No. 12 (1958)

pp. 1244-1248 [Ch 58d]

[Ch 59]

[Ch 63] [Ch 64] [Ch 69]

Remarks on my paper "The Jacobian variety of an algebraic curve," Amer. J. Math. 80 (1958) pp. 238-240 On the connectedness theorem in algebraic geometry, Amer. J. Math. 81 No. 4 (1959) pp. 1033-1074 On the real traces of analytic varieties, Am. J. Math. 85 No.4 (1963) pp. 723-733 On the unmixedness theorem, Am. J. Math. 86 (1964) pp. 799-822 On meromorphic maps of algebraic varieties, Ann. Math. 89 No. 2 (1969) pp. 391-403

xix

XX

[Ch 79]

[Ch 85]

[Ch 86] [Ch 92]

CHOW'S BIBLIOGRAPHY

On the algebraicity of certain ringed spaces, Amer. J. Math. 101 No. 2 (1979) pp. 364-379 Correction: "On the algebraicity of certain ringed spaces" [Ch 79], Amer. J. Math. 107 No. 3 (1985) pp. 759-760 Formal functions on homogeneous spaces, Invent. Math. 86 (1986) pp. 115-130 Shiing Shen Chern as friend and mathematieian, a reminiscence on the occasion of his 80th birthday. Chern~a great geometer of the twentieth century, Internat. Press, Hong Kong (1992) pp. 79-87

Selected Titles in This Series (Continued from the front of this publication)

179 Yoshiaki Maeda, Hideki Omoro, and Alan Weinstein, Editors, Symplectic geometry and quantization, 1994

178 Hel€me Barcelo and Gil Kalai, Editors, Jerusalem Combinatorics '93, 1994 177 Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr.,

Editors, Representation theory and analysis on homogeneous spaces, 1994 176 David Ballard, Foundational aspects of "non"standard mathematics, 1994 175 Paul J. Sally, Jr., Moshe Flato, James Lepowsky, Nicolai Reshetikhin,

and Gregg J. Zuckerman, Editors, Mathematical aspects of conformal and topological field theories and quantum groups, 1994

174 Nancy Childress and John W. Jones, Editors, Arithmetic geometry, 1994 173 Robert Brooks, Carolyn Gordon, and Peter Perry, Editors, Geometry of the

spectrum, 1994 172 Peter E. Kloeden and Kenneth J. Palmer, Editors, Chaotic numerics, 1994 171 Riidiger Gobel, Paul Hill, and Wolfgang Liebert, Editors, Abelian group theory

and related topics, 1994 170 John K. Beem and Krishan L. Duggal, Editors, Differential geometry and

mathematical physics, 1994 169 William Abikoff, Joan S. Birman, and Kathryn Kuiken, Editors, The

mathematical legacy of Wilhelm Magnus, 1994 168 Gary L. Mullen and Peter Jau-Shyong Shiue, Editors, Finite fields: Theory,

applications, and algorithms, 1994 167 Robert S. Doran, Editor, C*-algebras: 1943-1993, 1994 166 George E. Andrews, David M. Bressoud, and L. Alayne Parson, Editors, The

Rademacher legacy to mathematics, 1994 165 Barry Mazur and Glenn Stevens, Editors, p-adic monodromy and the Birch and

Swinnerton-Dyer conjecture, 1994 164 Cameron Gordon, Yoav Moriah, and Bronislaw Wajnryb, Editors, Geometric

topology, 1994 163 Zhong-Ci Shi and Chung-Chun Yang, Editors, Computational mathematics in

China, 1994 162 Ciro Ciliberto, E. Laura Livorni, and Andrew J. Sommese, Editors,

Classification of algebraic varieties, 1994 161 Paul A. Schweitzer, S. J., Steven Hurder, Nathan Moreira dos Santos, and Jose

Luis Arraut, Editors, Differential topology, foliations, and group actions, 1994 160 Niky Kamran and Peter J. Olver, Editors, Lie algebras, cohomology, and new

applications to quantum mechanics, 1994 159 William J. Heinzer, Craig L. Huneke, and Judith D. Sally, Editors,

Commutative algebra: Syzygies, multiplicities, and birational algebra, 1994 158 Eric M. Friedlander and Mark E. Mahowald, Editors, Topology and representation

theory, 1994 157 Alfio Quarteroni, Jacques Periaux, Yuri A. Kuznetsov, and Olof B. Widlund,

Editors, Domain decomposition methods in science and engineering, 1994 156 Steven R. Givant, The structure of relation algebras generated by relativizations, 1994 155 William B. Jacob, Tsit-Yuen Lam, and Robert 0. Robson, Editors, Recent

advances in real algebraic geometry and quadratic forms, 1994 154 Michael Eastwood, Joseph Wolf, and Roger Zierau, Editors, The Penrose

transform and analytic cohomology in representation theory, 1993 153 RichardS. Elman, Murray M. Schacher, and V. S. Varadarajan, Editors, Linear

algebraic groups and their representations, 1993 152 Christopher K. McCord, Editor, Nielsen theory and dynamical systems, 1993 151 Matatyahu Rubin, The reconstruction of trees from their automorphism groups, 1993

(See the AMS catalog for earlier titles)

Birational Algebraic Geometry Yujiro Kawamata and Vyacheslav V. Shokurov, Editors

This book presents proceedings from the Japan-U.S. Mathematics Institute (JAMI) Conference on Birational Algebraic Geometry in Memory of Wei-Liang Chow, held at the Johns Hopkins University in Baltimore in April 1996.

These proceedings bring to light the many directions in which birational al-gebraic geometry is headed. Featured are problems on special models , such as Fanos and their fibrations, adjunctions and subadjunction formuli, projectivity and projective embeddings, and more.

Some papers reflect the very frontiers of this rapidly developing area of math-ematics. Therefore, in these cases, only directions are given without complete explanations or proofs.

ISBN 0-8218-0769-2

9 780821 807699