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Functions of Several Complex Variables and Their Singularities Functions of Several Complex Variables and Their Singularities Wolfgang Ebeling Translated by Philip G. Spain Graduate Studies in Mathematics Volume 83 .•S%'3SL"?|| American Mathematical Society s^s^^v Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz Originally published in the German language by Friedr. Vieweg & Sohn Verlag, D-65189 Wiesbaden, Germany, as "Wolfgang Ebeling: Funktionentheorie, Differentialtopologie und Singularitaten. 1. Auflage (1st edition)". © Friedr. Vieweg & Sohn Verlag | GWV Fachverlage GmbH, Wiesbaden, 2001 Translated by Philip G. Spain 2000 Mathematics Subject Classification. Primary 32-01; Secondary 32S10, 32S55, 58K40, 58K60. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-83 Library of Congress Cataloging-in-Publication Data Ebeling, Wolfgang. [Funktionentheorie, differentialtopologie und singularitaten. English] Functions of several complex variables and their singularities / Wolfgang Ebeling ; translated by Philip Spain. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 83) Includes bibliographical references and index. ISBN 0-8218-3319-7 (alk. paper) 1. Functions of several complex variables. 2. Singularities (Mathematics) I. Title. QA331.E27 2007 515/.94—dc22 2007060745 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 a chapter 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 is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 Contents Foreword to the English translation ix Introduction xi List of figures xiii List of tables xvii Chapter 1. Riemann surfaces 1 §1.1. Riemann surfaces 1 §1.2. Homotopy of paths, fundamental groups 9 §1.3. Coverings 13 §1.4. Analytic continuation 24 §1.5. Branched meromorphic continuation 29 §1.6. The Riemann surface of an algebraic function 33 §1.7. Puiseux expansion 40 §1.8. The Riemann sphere 41 Chapter 2. Holomorphic functions of several variables 43 §2.1. Holomorphic functions of several variables 43 §2.2. Holomorphic maps and the implicit function theorem 57 §2.3. Local rings of holomorphic functions 60 §2.4. The Weierstrass preparation theorem 63 §2.5. Analytic sets 74 §2.6. Analytic set germs 76 §2.7. Regular and singular points of analytic sets 84 VI Contents §2.8. Map germs and homomorphisms of analytic algebras 89 §2.9. The generalized Weierstrass preparation theorem 96 §2.10. The dimension of an analytic set germ 101 §2.11. Elimination theory for analytic sets 109 Chapter 3. Isolated singularities of holomorphic functions 113 §3.1. Differentiable manifolds 113 §3.2. Tangent bundles and vector fields 119 §3.3. Transversality 125 §3.4. Lie groups 127 §3.5. Complex manifolds 134 §3.6. Isolated critical points 140 §3.7. The universal unfolding 144 §3.8. Modifications 149 §3.9. Finitely determined function germs 158 §3.10. Classification of simple singularities 165 §3.11. Real morsifications of the simple curve singularities 171 Chapter 4. Fundamentals of differential topology 181 §4.1. Differentiable manifolds with boundary 181 §4.2. Riemannian metric and orientation 183 §4.3. The Ehresmann fibration theorem 186 §4.4. The holonomy group of a differentiable fiber bundle 189 §4.5. Singular homology groups 194 §4.6. Intersection numbers 200 §4.7. Linking numbers 209 §4.8. The braid group 211 §4.9. The homotopy sequence of a differentiable fiber bundle 214 Chapter 5. Topology of singularities 223 §5.1. Monodromy and variation 223 §5.2. Monodromy group and vanishing cycles 226 §5.3. The Picard-Lefschetz theorem 229 §5.4. The Milnor fibration 238 §5.5. Intersection matrix and Coxeter-Dynkin diagram 249 §5.6. Classical monodromy, variation, and the Seifert form 252 §5.7. The action of the braid group 259 Contents vii §5.8. Monodromy group and vanishing lattice 269 §5.9. Deformation 277 §5.10. Polar curves and Coxeter-Dynkin diagrams 283 §5.11. Unimodal singularities 292 §5.12. The monodromy groups of the isolated hypersurface singularities 298 Bibliography 303 Index 307 Foreword to the English translation The German title of the book is "Funktionentheorie, Differentialtopologie und Singularitaten". The book is an introduction to the theory of functions of several complex variables and their singularities, with special emphasis on topological aspects. Its aim is to guide the reader from the fundamentals to more advanced topics of recent research. It originated from courses given by the author to German mathematics students at the University of Hanover. I am very happy that the AMS has provided an English edition of my book. I am grateful to Edward Dunne, the editor of the book program, for his efforts. My particular thanks go to Philip Spain, who translated this book into English. He has done a very good job. I have taken the opportunity to make some corrections and improve• ments in the text. I am grateful to Theo de Jong and Helmut Koditz for their comments and suggestions for improvement. Hanover, January 2007 Wolfgang Ebeling IX Introduction The study of singularities of analytic functions can be considered as a sub- area of the theory of functions of several complex variables and of alge• braic/analytic geometry. It has in the meantime, together with the theory of singularities of differentiable mappings, developed into an independent subject, singularity theory. Through its connections with very many other mathematical areas and applications to natural and economic sciences and in technology (for example, under the heading 'catastrophe theory') this theory has aroused great interest. The particular appeal, but also its par• ticular difficulty, lies in the fact that deep results and methods from various branches of mathematics come into play here. The aim of this book is to present the foundations of the theory of func• tions of several complex variables and on this basis to develop the fundamen• tal concepts of the theory of isolated singularities of holomorphic functions systematically. It is derived from lectures given by the author to mathe• matics students in their third and fourth year to introduce them to current research questions in the area of the theory of functions of several variables. The book has its genesis in this. As prerequisites we assume only an intro• ductory knowledge of the theory of functions of a single complex variable and of algebra, such as students will normally acquire in their first two years of study. The first two chapters correspond to a continuation of the course on complex analysis and deal with Riemann surfaces and the theory of func• tions of several complex variables. They also present an introduction to local complex geometry. In the third chapter the results will be applied to defor• mation and classification of isolated singularities of holomorphic functions. These three chapters have grown from notes for the author's lectures on Riemann surfaces and the theory of functions of several complex variables XI Xll Introduction delivered in Hanover in the winter semester of 1998/1999 and the summer semester of 1999. Parts of these notes go back to similar courses given in the winter semester of 1992/1993 and the summer semester of 1993. The rest of the book deals with the topological study of these singulari• ties begun in the now classical book of J. Milnor [Mil68]. Picard-Lefschetz theory is an important tool and can be viewed as a complex version of Morse theory. It is expounded at the beginning of the second volume of the ex• tensive two-volume standard work of V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko [AGV85, AGV88]. These books assume considerable prior knowledge. We offer an introduction to this theory in the last two chapters of the present book. In the fourth chapter we first present the necessary foundations of algebraic and of differential topology. The fifth chapter introduces the topological study of singularities. It rests in part on [AGV88, Part I. The topological structure of isolated critical points of functions]. At the end of this chapter there is a survey of topical results, some presented without proof. The last two chapters are based on a course on singularities delivered by the author in Hanover in the winter semester of 1993/1994. This book can be used for a course on functions of several complex vari• ables, an introductory course on differential topology, or for a special course or seminar on an introduction to singularity theory. The first two chapters would be suitable for a further course on functions of several complex vari• ables. The beginning of §1.1, §1.2, and the first four sections of Chapter 3 and Chapter 4 treat themes from differential topology and can be read in• dependently of the rest of the book: they can therefore serve as the basis of an introductory course on differential topology. Chapter 3 and Chapter 5 can be used as reading for a seminar on Introductory singularity theory, with reference back to the results of the previous chapters according to the state of knowledge of the participants.
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