Embed Size (px)
Citation preview
Several Complex Variables and Complex Geometry
http://dx.doi.org/10.1090/pspum/052.2
Proceedings of Symposia in
PURE MATHEMATICS
Volume 52, Part 2
Several Complex Variables and Complex Geometry
Eric Bedford John P. D'Angelo Robert E. Greene Steven G. Krantz Editors
^ T T P H T O Z M H N ^
^ / / ? ^ ^ ^ \ G American Mathematical Society v Providence, Rhode Island
PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON SEVERAL COMPLEX VARIABLES AND COMPLEX GEOMETRY
HELD AT THE UNIVERSITY OF CALIFORNIA, SANTA CRUZ SANTA CRUZ, CALIFORNIA
JULY 10-30, 1989
with the support of the National Science Foundation Grant DMS-8814802
1980 Mathematics Subject Classification (1985 Revision). Primary 32A, 32D, 32E, 32F, 32H (Part 1)
32B, 32C, 32G, 32H, 32J, 32K, 32L, 32M, 53C (Part 2) 35N15, 32F20, 53B, 32A, 32F, 32C (Part 3)
Library of Congress Cataloging-in-Publication Data
Summer Research Institute on Several Complex Variables and Complex Geometry (1989: University of California, Santa Cruz)
Several complex variables and complex geometry/[edited by] Eric Bedford . . . [et al.]. p. cm.—(Proceedings of symposia in pure mathematics, ISSN 0082-0717; v. 52) "Proceedings of the Summer Research Institute on Several Complex Variables and Com
plex Geometry, held at the University of California, Santa Cruz, Santa Cruz, California, July 10-30, 1989"—T.p. verso.
Includes bibliographical references. 1. Functions of several complex variables—Congresses. 2. Geometry, Differential—Con
gresses. I. Bedford, Eric, 1947- . II. American Mathematical Society. III. Title. IV. Series. QA331.7.S86 1989 91-11227 515'.94—dc20 CIP ISBN 0-8218-1489-3 (part 1) ISBN 0-8218-1490-7 (part 2) ISBN 0-8218-1491-5 (part 3) ISBN 0-8218-1488-5 (set: alk. paper)
COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248.
The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale.
Copyright ©1991 by the American Mathematical Society. All rights reserved. Printed in the United States of America.
The American Mathematical Society retains all rights except those granted to the United States Government.
The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @
This publication was typeset using AMS-TJJX, the American Mathematical Society's TEX macro system.
10 9 8 7 6 5 4 3 2 1 96 95 94 93 92 91
Contents
Preface xiii
PARTI
Uniqueness for the Dirichlet Problem for Harmonic Maps from the Annulus into the Space of Planar Discs DAVID E. BARRETT 1
CR Maps Between Hypersurfaces in Cn
STEVE BELL 13
Bounds for the Degrees in Polynomial Equations CARLOS A. BERENSTEIN AND ALAIN YGER 23
Lagrange Interpolants for Entire Functions on &n
THOMAS BLOOM 29
An Inequality for Analytic Functions URBAN CEGRELL 37
Some Solutions of the Beltrami Equation with ||//||oo = 1 J. A. CIMA AND W. R. DERRICK 41
On the Problem of the Linearization of the Stability Group of a Real-Analytic Hypersurface V. EZHOV 45
Mappings of Strongly Pseudoconvex Cauchy-Riemann Manifolds FRANC FORSTNERIC 59
Rational Curves and Extensions of Holomorphic Mappings S. M. IVASHKOVICH 93
Applications of C" Capacities JACOB KOREVAAR 105
vi CONTENTS
Convexity in Complex Analysis STEVEN G. KRANTZ 119
Solutions of d2u/dzdz = e2u and Analytic Multivalued Functions DONNA KUMAGAI 13 9
Local Versus Global Convexity of Pseudoconvex Domains ALANNOELL 145
The Scaling Method and Holomorphic Mappings SERGEY PINCHUK 151
Plurisubharmonic Functions as Solutions of Variational Problems EVGENY A. POLETSKY 163
On Complex Analysis in Tube Cones A. G. SERGEEV 173
Separately Meromorphic Functions and Separately Holomorphic Mappings BERNARD SHIFFMAN 191
Some Aspects of Weakly Pseudoconvex Domains NESSIM SIBONY 199
Unions Minimales de «-Plans Reels d'Enveloppe Egale a Cn
PASCAL J. THOMAS 233
Separately Subharmonic Functions: When are They Subharmonic JAN WIEGERINCK AND REIN ZEINSTRA 245
Functions of Bounded Mean Oscillation in the Bergman Metric on Bounded Symmetric Domains KEHEZHU 251
PART 2
The Geometry of Complex Manifolds: An Overview ROBERT E. GREENE 1
Angular Derivatives in Strongly Pseudoconvex Domains MARCO ABATE 23
CONTENTS vii
Degeneracy Points of the Kobayashi Pseudodistances on Complex Manifolds YUKINOBU ADACHI AND MASAKAZU SUZUKI 41
On the Construction of the Moduli Space for Strongly Pseudoconvex Domains TAKAO AKAHORI 53
On the Holomorphic Sectional Curvature of Complete Domains in Cn
that are not Stein ANDREW BALAS 59
Normal Forms for Convex Domains J. BLAND AND T. DUCHAMP 65
Homogeneous Vector Bundles and Families of Calabi-Yau Threefolds. II ClPRIAN BORCEA 83
Holomorphic Morse Inequalities JEAN PIERRE DEMAILLY 93
Some Problems PIERRE DOLBEAULT 115
Invariants of CR Densities MICHAEL G. EASTWOOD AND C. ROBIN GRAHAM 117
Convex Symplectic Manifolds YAKOV ELIASHBERG AND MIKHAEL GROMOV 135
Simple Partial Connections and the Einstein Condition H. R. FISCHER AND R. J. FISHER, JR. 163
Applications of Affine Geometry to Geometric Function Theory in Several Complex Variables Part I. Convergent Rescalings and Intrinsic Quasi-Isometric Structure SIDNEY FRANKEL 18 3
A Lie Algebra Character and Kahler-Einstein Metrics of Positive Scalar Curvature AKITO FUTAKI 209
viii CONTENTS
On the Ends of Complex Manifolds Homogeneous Under a Lie Group BRUCE GILLIGAN 217
Determinant Line Bundles for Hermitian Foliations and a Generalized Quillen Metric JAMES F. GLAZEBROOK AND FRANZ W. KAMBER 225
Sharp Constants for the Koebe Theorem and for Estimates of Intrinsic Metrics on Convex Domains IAN GRAHAM 233
Rigid Immersions of G-Structures and Analyticity of CR Mappings into Spheres CHONG-KYU HAN 239
Optimal Analytic Disks J. WILLIAM HELTON AND ORLANDO MERINO 251
Some Remarks on the Product Property for Invariant Pseudometrics MAREK JARNICKI AND PETER PFLUG 263
Smoothing of Currents and Moisezon Manifolds SHANYU JI 273
Biholomorphic Mappings Between Quasicircular Domains in Cn
KANG-TAE KIM 283
Characterizations of Certain Weakly Pseudoconvex Domains in Cn from the Viewpoint of Biholomorphic Automorphism Groups AKIO KODAMA 291
Complete Ricci-Flat Kahler Metrics on Cn Need Not Be Flat CLAUDE LEBRUN 297
On Meromorphic Maps into Varieties of Log-General Type STEVEN SHIN-YI LU 305
Aspects of Kahler Geometry on Arithmetic Varieties NGAIMING MOK 335
Moduli Spaces of Holomorphic Mappings into Hyperbolic Spaces and its Applications J. NOGUCHI 397
CONTENTS ix
The Analytic Radon Transform SALOMON OFMAN 403
Applications of the d Technique in L2 Hodge Theory on Complete Kahler Manifolds TAKEO OHSAWA 413
Bounds on the Derivatives of Holomorphic Endomorphisms MARIUS OVERHOLT 427
Monge-Ampere Functions with Large Center GIORGIO PATRIZIO AND PIT-MANN WONG 435
A Characterization of Affine Varieties MATHIAS PETERNELL 449
Compactifications of Cn: A Survey THOMAS PETERNELL AND MICHAEL SCHNEIDER 455
An Infinite-Dimensional Point of View on the Weil-Petersson Metric T. RATIU AND A. TODOROV 467
The Cartan Conjecture for Moving Targets MIN Ru AND WILHELM STOLL 477
On Kollar's Conjecture MORIHIKO SAITO 509
An Introduction to L2-Cohomology LESLIE SAPER AND STEVEN ZUCKER 519
On the Compactification Problems for Stein 3-Folds Vo VAN TAN 535
On One of Calabi's Problems GANG TIAN 543
Multi-Dimensional Analytic Structures and Uniform Algebras T. TONEV 557
The d d -Equation on Complex Spaces J. VAROUCHAS 563
X CONTENTS
Fixed Points of Holomorphic Mappings in a Bounded Convex Domain inC" JEAN-PIERRE VIGU£ 579
Holomorphic Functions Taking Their Values in a #-Space and the Cauchy-Fantappie Formula L. WAELBROECK 583
Schwarz's Lemma and Hermitian Manifolds with Constant Holomorphic Curvature B. WONG 593
Polynomial Functions on Complete Kahler Manifolds H. Wu 601
The Chern Numbers of Projective Algebraic Hypersurfaces KICHOON YANG 611
A Review of Complex Differential Geometry SHING-TUNG YAU 619
PART 3
Minimality and the Extension of Functions from Generic Manifolds M. S. BAOUENDI AND LINDA PREISS ROTHSCHILD 1
Holomorphic Mappings of Real Analytic Hypersurfaces M. S. BAOUENDI AND LINDA PREISS ROTHSCHILD 15
Extension of CR Structures DAVID CATLIN 27
%?v Regularities for NIS Operators on Weakly Pseudo-Convex Domains of Finite type in C2
DER-CHEN E. CHANG AND STEVEN G. KRANTZ 35
Global Regularity of the d-Neumann Problem in Dimension Two SO-CHIN CHEN 55
On the di, Equation for Three-Dimensional CR Manifolds MICHAEL CHRIST 63
CONTENTS XI
Necessary Conditions for the Local Solvability of the Tangential CR Equations PAULO CORDARO AND FRANCOIS TREVES 83
Operator Theory on Reinhardt Domains RATJL CURTO 93
Finite Type and the Intersection of Real and Complex Subvarieties JOHN P. D'ANGELO 103
Local Analyticity for the d-Neumann Problem and for Ob in Some Model Domains without Maximal Estimates MAKHLOUF DERRIDJ AND DAVID S. TARTAKOFF 119
On Lp Estimates for 5 JOHN ERIK FORNAESS AND NESSIM SIBONY 129
Approximate Projections and Analytic Pseudolocality of the Szego Projection on Domains in C2
DARYL GELLER 16 5
On an Example of a Subelliptic Boundary Value Problem PENGFEI GUAN 173
Degenerate Surfaces in C2
GARY A. HARRIS 179
Counterexamples to Newlander-Nirenberg up to the Boundary C. DENSONHILL 191
Finite Type Analytic Partial Differential Operators and Analytic Hypoellipticity A. ALEXANDROU HIMONAS 199
Singularities of Harmonic Functions in Cn
D. KHAVINSON 207
A Study of the Bergman Projection in Certain Hartogs Domains CHRISTER O. KISELMAN 219
The Frame Bundles of CR Structures and the Bergman Kernel (I) MASATAKE KURANISHI 233
xii CONTENTS
Resolution du ~d\j a Support Compact et Phenomene de Hartogs-Bochner dans les Varietes CR CHRISTINE LAURENT-THIEBAUT 239
On the Regularity of Coanalytic Toeplitz Operators EWALIGOCKA 251
Obstructions to Embedding of Real (2n - 1)-Dimensional Compact CR Manifolds in Cw+1
HlNG-SUN LUK AND STEPHEN S.-T. YAU 261
On Sharp Holder Estimates for the Solutions of the ^-Equations JEFFERY D. MCNEAL 277
Equivalence of Analytic and Plurisubharmonic Phragmen-Lindelof Conditions REINHOLD MEISE, B. A. TAYLOR, AND DIETMAR VOGT 287
The Cauchy-Riemann Equations in Convex Domains JOHN C. POLKING 309
Examples of Smooth CR Structures with only Nonsmooth CR Functions JEAN-PIERRE ROSAY 323
Noncompactness of the d-Neumann Problem and Toeplitz C*-Algebras NORBERTO SALINAS 329
Local Solvability and Estimates for ~db on CR Manifolds MEI-CHI SHAW 335
Rigid Hypersurfaces in C2
NANCY K. STANTON 347
The Integrability Problem for CR Vector Bundles S. M. WEBSTER 355
Preface
The American Mathematical Society held its thirty-seventh annual Summer Research Institute at the University of California at Santa Cruz, Santa Cruz, California, from July 10 to July 30, 1989. The focus of this institute was several complex variables and complex geometry. The members of the AMS Committee on Summer Institutes that selected this program were Eric Friedlander, Steven L. Kleiman, Paul H. Rabinowitz, Thomas C. Spencer, Robert B. Warfield, Jr., and John Wermer (chair). The institute was supported in part by a grant from the National Science Foundation. The Organizing committee for the institute consisted of Eric Bedford, John D'Angelo, Robert E. Greene, and Steven G. Krantz (chair). All of the organizers edited these proceedings.
The institute explored recent developments in the geometry and function theory of several complex variables. An attempt was made to create a dialogue among the different methodologies in the subject: differential geometry, algebraic geometry, partial differential equations, harmonic analysis, and classical methods. Each week there were two series of several connected lectures (called the Principal Lectures) representing important directions in the field of emphasis for that week. These were intermixed with single hour lectures. Both the connected series of talks and the hour lectures were presented in the mornings and early afternoons. The last three hours of every day consisted of two parallel sessions of briefer seminar lectures.
Principal lectures
Week One Nessim Sibony, Weakly Pseudoconvex Domains Sergey Pinchuk, The Scaling Method and Holomorphic Mappings
Week Two Ngaiming Mok, Problems in Kahler Geometry Related to Hermitian
{Locally) Symmetric Spaces Jean Pierre Demailly, Holomorphic Morse Inequalities
Week Three David Catlin, Extensions ofCR Structures John Erik Fornaess, Lp Estimates for d
xii i
XIV PREFACE
Hour lectures Steven R. Bell, CR Maps and Local Regularity Jean Martinet, Wild Cauchy Theory and Applications Berit Stensones, Aspects of Envelopes ofHolomorphy David Barrett, Uniqueness for a Nonlinear Dirichlet Problem Connected
with One Dimensional Function Theory Jean-Pierre Rosay, About the Embeddability ofCR Structures B. A. Taylor, Equivalence of Analytic and Plurisubharmonic
Phragmen-Lindelof Conditions Franc Forstneric, Mappings ofCR Manifolds Vladimir Ezhov, The Problem of Linearization of the Stability Group S. T. Yau, Uniformization Sergeii Ivashkovitch, Rational Curves and Extension of Holomorphic Maps Jung Hsi Wu, Holomorphic Functions of Polynomial Growth on Kahler
Manifolds Michael Schneider, On Two Conjectures ofHartshorne Steven Zucker, A Survey of L Cohomology Leslie Saper, L Cohomology of Algebraic Varieties Y. T. Siu, Global Nondeformability of Compact Hermitian Symmetric
Manifolds Takeo Ohsawa, Convexity, Concavity, and L2 Cohomology on Noncompact
Manifolds Akito Futaki, A Lie Algebra Character and Kahler-Einstein Metrics of Pos
itive Scalar Curvature Edward Bierstone, Local Resolution of Singularities James Carlson, Harmonic Mappings on Kahler Manifolds Gang Tian, On One ofCalabVs Problems Linda Rothschild, CR Functions on Manifolds of Higher Codimension Sidney Webster, The Integrability Problem for CR Vector Bundles M. Salah Baouendi, General Reflection Principles J. J. Kohn, Pseudoconvex CR Manifolds with Diagonalizable Levi Form Francois Treves, On the Local Solvability of the Tangential Cauchy-
Riemann Equations Yakov Eliashberg, Topological Characterization of Stein Manifolds of
Dimension > 2 F. Michael Christ, Estimates for db on Finite Type CR Manifolds Armen Sergeev, Complex Analysis in Tube Cones Elias M. Stein, Analysis on Finite Type Domains
The purpose of the papers in these volumes is to exhibit and describe the current state of the field. The papers are organized according to the week in which the corresponding lectures were presented, and are given in alphabetical order. The first week was devoted to function theory, the second to complex geometry, and the third to analysis and partial differential equations.
PREFACE xv
The University of California at Santa Cruz provided pleasant facilities and a beautiful environment in which to conduct our sessions. Wayne Drady of the American Mathematical Society made the entire event run both smoothly and harmoniously. The American Mathematical Society was a great help at every stage of the planning.
We dedicate these volumes to the memory of Jean Martinet.
E. Bedford J. P. D'Angelo R. E. Greene S. G. Krantz
