CR-Geometry and Overdetermined Systems

Transcription

1 CR-Geometry and Overdetermined Systems

2 ADVANCED STUDIES IN PURE MATHEMATICS 25 Chief Editor: Tadao Oda (Tohoku University) CR-Geometry and Overdetermined Systems Edited by Takao Akahori (Himeji Institute of Technology) Gen Komatsu (Osaka University) Kimio Miyajima (Kagoshima University) Makoto Namba (Osaka University) and Keizo Yamaguchi (Hokkaido University) Published for the Mathematical Society of Japan by KINOKUNIYA COMPANY LTD. TOKYO, JAPAN

3 This book was typeset by AMS-TEX and AMS-LATEX, the TEX macro systems of the American Mathematical Society, together with the style files aspm.sty and aspmfm.sty for AMS-TEX written by Dr. Chiaki Tsukamoto and aspmproc.sty for AMS-LATEX written by Dr. Akihiro Munemasa. TEX isatrademark of the American Mathematical Society. c 1997 by the Mathematical Society of Japan. All rights reserved. No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. Edited by the Mathematical Society of Japan. Published for the Mathematical Society of Japan and distributed in Japan by Kinokuniya Company Ltd., Tokyo, Japan. Distributed outside Japan by the American Mathematical Society. ISBN X 1991 Mathematics Subject Classification. Primary 32Fxx, 32Gxx; Secondary 33Cxx, 35Mxx, 53Cxx, 58Gxx. Advanced Studies in Pure Mathematics 25 Hitoshi Arai Mitsuru Ikawa (Osaka Univ.) Yujiro Kawamata Shigeru Mukai (Nagoya Univ.) Takushiro Ochiai Toshikazu Sunada Chief Editor Tadao Oda (Tohoku University) Editorial Board of the Series Eiichi Bannai (Kyushu Univ.) Masaki Kashiwara Tetsuji Miwa Seiki Nishikawa Tadao Oda Yoichiro Takahashi PRINTED IN JAPAN by Tokyo Shoseki Printing Co., Ltd. Yasutaka Ihara Kazuya Kato Shigeyuki Morita Junjiro Noguchi (Tokyo Inst. of Tech.) Toshio Oshima Toshiyuki Tanisaki (Hiroshima Univ.)

4 Preface This volume presents contributions focused on recent developments of CR geometry and overdetermined systems. Some of the papers are based on the lectures delivered at a conference of the same title held at Osaka, Japan, from December 19 to 21, 1994, on the occasion of Kuranishi s 70th birthday. The notion of CR manifold is an abstraction of a smooth boundary of a complex manifold or a complex space equipped with the tangential Cauchy-Riemann operator b, where a formal integrability condition is imposed as in the case of an almost complex structure to be a complex structure. When the strict pseudoconvexity is assumed, one can employ differential geometric formalism and the method of harmonic integrals for the b complex analogous to the -Neumann problem. How does the CR structure on the boundary determine the complex structure inside? This is a central theme of CR geometry. A fundamental question is the embeddability of a CR manifold M as a real hypersurface of a complex space. An affirmative answer for the local embeddability was given by Kuranishi under the assumption that dim R M is not too small, indeed, dim R M 7issufficient. (dim R M must be odd.) The case dim R M =3 is exceptional, and most of the CR structures on M are not embeddable even when M is compact. In the early 1970s, Kuranishi proposed to develop a deformation theory of isolated singularities via that of embeddable CR structures. His idea was presented in 1975 by series of lectures at the AMS summer institute, Williamstown, and at RIMS, Kyoto University. Since then, great progress was made of the theory and applications of CR geometry, synchronously with attempts to realize Kuranishi s idea. This volume reports on such progress and related topics as follows. Methods of studying isolated singularities are developed since 1975, and the article by Ohsawa overviews such development. To investigate the deformations of a three dimensional CR manifold, one must consider the spaces of embeddable and non-embeddable CR structures, and this is done in the papers by Bland, Epstein and Lempert. The local embedding problem for 5-dimensional CR manifolds is still open, and Webster investigates this through a model problem. The paper by Luk and Stephen Yau discusses problems related to the minimal embedding dimension of a compact CR manifold in the Euclidean space. Since the work of E. Cartan, the method of prolongation has been successfully used in CR geometry. The paper by Han and Yoo determines the freedom of the CR mappings via the prolongation, while Veloso discusses

5 the Cartan connection of CR structure through the general theory of prolongation. The weakly pseudoconvex case is difficult due to the lack of differential geometric formalism, and the paper by Stanton deals with this case by studying infinitesimal CR automorphisms. The higher codimensional CR geometry is also difficult, and the paper by Garrity and Mizner discuss invariants of the Levi form in this case. There is also a variety of interesting contributions on related topics. The article by DeTurk and Goldschmidt considers problems of seeking a Riemannian metric with the Ricci curvature prescribed. The paper by Gasqui and Goldschmidt deals with a theorem on the spectral rigidity. The paper by Glazebrook and Sundararaman is concerned with the deformations of self-dual bundles. The article by Phong and Silvotti should be useful to mathematicians interested in the conformal field theory. The paper by Sasaki, Yamaguchi and Yoshida is concerned with a conjecture on hypergeometric functions of several variables. Umemura gives a new insight to the infinite dimensional differential Galois theory. Three survey lecture notes are included for the reader s convenience. The surveys by Akahori and by Miyajima are concerned with recent development on the deformations of isolated singularities of higher dimension via those of CR structures. The survey by Hirachi and Komatsu are on the invariant theory of the Bergman kernel initiated by Fefferman. These are attempts to understand the relation between the CR structure on the boundary and the local biholomorphic geometry inside. Acknowledgment We would like to thank those who helped us in many ways to prepare this volume. Especially, we would like to thank the editors of the Advanced Studies in Pure Mathematics series, in particular, Professor Tadao Oda for giving us valuable advice, and Professor Shigeru Mukai for providing financial support for our activities related to the organization of this volume. We also thank the referees who gave valuable comments to contributors to this volume. Finally, we would like to thank the secretaries of the editorial office of the Advanced Studies in Pure Mathematics series. M. Namba and K. Yamaguchi On behalf of the editors All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere.

6 CONTENTS Takao Akahori Deformation theory of CR-structures and its application to deformations of isolated singularities I 1 John Bland and Thomas Duchamp Deformation theory for the hyperplane line bundle on P 1 41 Dennis DeTurck and Hubert Goldschmidt Aspects of prescribing Ricci curvature 60 Charles L. Epstein Deformations of singularities, complex manifolds and CR-structures 85 Thomas Garrity and Robert Mizner Vector-valued forms and CR geometry 110 Jacques Gasqui and Hubert Goldschmidt The infinitesimal spectral rigidity of the real Grassmannians of rank two 122 James F. Glazebrook and Duraiswamy Sundararaman On deformations of self-dual vector bundles over quaternionic manifolds 141 Chong-Kyu Han and Jae-Nyun Yoo Amethod of prolongation of tangential Cauchy-Riemann equations 158 Kengo Hirachi and Gen Komatsu Invariant theory of the Bergman kernel 167 László Lempert Spaces of Cauchy-Riemann manifolds 221 Hing Sun Luk and Stephen S. T. Yau Some remarks on compact strongly pseudoconvex CR manifolds 237 Kimio Miyajima Deformation theory of CR-structures and its application to deformations of isolated singularities II 247 Takeo Ohsawa Areport on isolated singularities by transcendental methods 276 Duong H. Phong and Roberto Silvotti The fusion matrix and the Verlinde loop operators in conformal field theory 285 Takeshi Sasaki, Keizo Yamaguchi and Masaaki Yoshida On the rigidity of differential systems modelled on Hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces 318

7 Nancy K. Stanton Infinitesimal CR automorphisms 355 Hiroshi Umemura Lie-Drach-Vessiot theory Infinite dimensional differential Galois theory 364 Jose M. Veloso Prolongation projection commutativity theorem 386 Sidney M. Webster A complex Frobenius problem 406