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L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Second Edition Springer Monographs in Mathematics Springer Monographs in Mathematics Takeo Ohsawa L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Second Edition Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto Pinto, Porto, Portugal Gabriella Pinzari, Napoli, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research. More information about this series at http://www.springer.com/series/3733 Takeo Ohsawa L2 Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Second Edition 123 Takeo Ohsawa Professor Emeritus Nagoya University Nagoya, Japan ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-4-431-56851-3 ISBN 978-4-431-56852-0 (eBook) https://doi.org/10.1007/978-4-431-56852-0 Library of Congress Control Number: 2018959147 © Springer Japan KK, part of Springer Nature 2015, 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Japan KK part of Springer Nature. The registered company address is: Shiroyama Trust Tower, 4-3-1 Toranomon, Minato-ku, Tokyo 105-6005, Japan Preface As in the study of complex analysis of one variable, the general theory of several complex variables has manifold aspects. First, it provides a firm ground for systematic studies of special functions such as elliptic functions, theta functions, and modular functions. The general theory plays a role of confirming the existence and uniqueness of functions with prescribed zeros and poles. Another aspect is to give an insight into the connection between two different fields of mathematics by understanding how the tools work. The theory of sheaves bridged analysis and topology in such a way. In the construction of this basic theory of several complex variables, a particularly important contribution was made by two mathematicians, Kiyoshi Oka (1901–1978) and Henri Cartan (1904–2008). The theory of Oka and Cartan is condensed in a statement that the first cohomology of coherent analytic sheaves over Cn is zero. On the other hand, the method of PDE (partial differential equations) had turned out to be essential in the existence of conformal mappings. By this approach, the function theory on Riemann surfaces as one-dimensional complex manifolds was explored by H. Weyl. Weyl’s method was developed on manifolds of higher dimension by K. Kodaira who generalized Riemann’s condition for Abelian varieties by establishing a differential geometric characterization of nonsingular projective algebraic varieties. This PDE method, based on the L2 estimates for the ∂¯-operator, was generalized by J. Kohn, L. Hörmander, A. Andreotti, and E. Vesentini. As a result, it enabled us to see the results of Oka and Cartan in a much higher resolution. In particular, based on such a refinement, existence theorems for holomorphic functions with L2 growth conditions have been obtained by Hörmander, H. Skoda, and others. The purpose of the present monograph is to report on some of the recent results in several complex variables obtained by the L2 method which can be regarded as a continuation of these works. Among various topics including complex geometry, the Bergman kernel, and holomorphic foliations, a special emphasis is put on the extension theorems and its applications. In this topic, highlighted are the recent developments after the solution of a long- standing open question of N. Suita. It is an inequality between the Bergman kernel and the logarithmic capacity on Riemann surfaces, which was first proved by Z. Błocki for plane domains. Q. Guan and X.-Y. Zhou proved generalized variants v vi Preface and characterized those surfaces on which the inequality is strict. Their work gave the author a decisive impetus to start writing a survey to cover these remarkable achievements. As a result, he could find an alternate proof of the inequality, based on hyperbolic geometry, which is presented in Chap. 3. However, the readers are recommended to have a glance at Chap. 4 first, where the questions on the Bergman kernels are described more systematically. (The author started to write the monograph from Chap. 4.) Since there have been a lot of subsequent progress concerning the materials in Chaps. 3 and 4 during the preparation of the manuscript, it soon became beyond the author’s ability to give a satisfactory account of the whole development. So he will be happy to have a chance in the future to revise and enlarge this rather brief monograph. Nagoya, Japan Takeo Ohsawa March 2015 Preface to the Second Edition Thanks to the goodwill of the publisher, the revision and enlargement have been realized. What made this edition possible was the recent remarkable activity after Błocki’s solution of Suita’s conjecture for plane domains. Among many corrections, the most important one is the replacement of an erroneous proof of Theorem 3.2 by the present one which is hopefully correct. The author is very grateful to Shigeharu Takayama for pointing out the mistake. Additions have been made to focus on the results which appeared in the past 3 years. Some of them are in Sect. 4.4.5 “Berndtsson–Lempert Theory and Beyond” and in the section “A History of Levi Flat Hypersurfaces”in5.3. Besides these, each chapter has been supplemented by a section titled “Notes and Remarks,” in which the author also tried to enhance the depth feeling of complex analysis and convey the atmosphere of several complex variables similar to searching for extraterrestrial intelligence since Hartogs and Oka. Nagoya, Japan Takeo Ohsawa April 2018 vii Contents 1 Basic Notions and Classical Results........................................ 1 1.1 Functions and Domains Over Cn ....................................... 2 1.1.1 Holomorphic Functions and Cauchy’s Formula............... 2 1.1.2 Weierstrass Preparation Theorem .............................. 4 1.1.3 Domains of Holomorphy and Plurisubharmonic Functions .. 7 1.2 Complex Manifolds and Convexity Notions ........................... 9 1.2.1 Complex Manifolds, Stein Manifolds and Holomorphic Convexity ....................................................... 10 1.2.2 Complex Exterior Derivatives and Levi Form................. 14 1.2.3 Pseudoconvex Manifolds and Oka–Grauert Theory .......... 16 1.3 Oka–Cartan Theory ..................................................... 19 1.3.1 Sheaves and Cohomology ...................................... 19 1.3.2 Coherent Sheaves, Complex Spaces, and Theorems A and B ............................................................ 25 1.3.3 Coherence of Direct Images and a Theorem of Andreotti and Grauert .......................................... 30 1.4 ∂¯-Equations on Manifolds .............................................. 31 1.4.1 Holomorphic Vector Bundles and ∂¯-Cohomology ............ 32 1.4.2 Cohomology with Compact Support........................... 36 1.4.3 Serre’s Duality Theorem ....................................... 38 1.4.4 Fiber Metric and L2 Spaces .................................... 41 1.5 Notes and Remarks ..................................................... 42 References ..................................................................... 45 2 Analyzing the L2 ∂¯-Cohomology ........................................... 47 2.1 Orthogonal Decompositions in Hilbert Spaces ........................ 47 2.1.1 Basics on Closed Operators ...................................
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