Paul M. Gauthier Lectures on Several Complex Variables
Paul M. Gauthier Départment de Mathématiques et de Statistique Université de Montréal Montreal, QC, Canada ISBN 978-3-319-11510-8 ISBN 978-3-319-11511-5 (ebook) DOI 10.1007/978-3-319-11511-5 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014951537 Mathematics Subject Classification (2010): 3201 Springer International Publishing Switzerland 2014 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface This is a slightly amplified English translation of a course given at Université de Montréal in 2004. Prerequisites for the course are functions of one complex variable and functions of several real variables and topology, all at the undergraduate level. A previous encounter with subharmonic functions in the complex plane would be helpful, but the reader may consult introductory texts on complex analysis for such material. We deliberately require the student to fill in many details, but we indicate precisely when this is the case. Some important theorems are stated as problems. These missing details and proofs are carefully orchestrated to be feasible and instructive in context, and the course in this format was a success. These notes give only a short and swift overview of concepts and naturally indicate my own preferences, so instructors could use these notes as an indicator and then build their lectures of their own liking, with additional examples and exercises. While all of the textbooks in the bibliography are excellent, I wish to point out the unusual nature of the book by Kaplan [7]. Kaplan s book is one of the rare books that presents several complex variables at the undergraduate level and yet manages to present a significant amount of important material. Moreover, it has many elementary exercises and excellent illustrations. While the present notes are aimed primarily at graduate students, more ambitious undergraduate students and research mathematicians whose specialty is not several complex variables may also benefit from these lecture notes. It is my hope that this introduction to complex analysis in several variables, though brief, will be sufficient to inspire the student who has gone through them to have the curiosity to attend many interesting colloquia and seminars that touch upon topics related to those herein presented. It is a pleasure to thank our friend Michael Range for helpful suggestions. Montreal, QC, Canada Paul M. Gauthier v
Contents 1 Introduction... 1 2 Basics... 3 3 Cauchy Integral Formula... 9 4 Sequences of Holomorphic Functions... 15 5 Series... 19 6 Zero Sets of Holomorphic Functions... 25 7 Holomorphic Mappings... 31 8 Plurisubharmonic Functions... 39 9 The Dirichlet Problem... 51 10 Uniform Approximation... 53 11 Complex Manifolds... 55 12 Examples of Manifolds... 59 12.1 Domains... 59 12.2 Submanifolds... 59 12.3 Product Manifolds and Matrices... 62 12.4 Lie Groups... 63 12.5 ProjectiveSpace... 64 12.6 GrassmannManifolds... 66 12.7 Tori... 68 12.8 The Quotient Manifold with Respect to an AutomorphismGroup... 70 12.9 SpreadManifolds... 74 13 Holomorphic Continuation... 77 13.1 Direct Holomorphic Continuation and Domains of Holomorphy.. 77 13.2 IndirectHolomorphicContinuationand RiemannDomains... 80 vii
viii Contents 14 The Tangent Space... 85 14.1 Hermitian Manifolds... 89 14.2 Symplectic Manifolds... 95 14.3 Almost ComplexManifolds... 97 15 Meromorphic Functions and Subvarieties... 101 15.1 MeromorphicFunctions... 101 15.2 Subvarieties... 104 References... 107 Index... 109