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Several Complex Variables and Complex Geometry Proceedings of Symposia in PURE MATHEMATICS http://dx.doi.org/10.1090/pspum/052.2 Several Complex Variables and Complex Geometry 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 ^TTPHTOZMHN ^ ^//?^^^\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 li• braries 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 publi• cation (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.
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