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

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 of an algebraic function 33 §1.7. Puiseux expansion 40 §1.8. The 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 101 §2.11. Elimination theory for analytic sets 109 Chapter 3. Isolated singularities of holomorphic functions 113 §3.1. Differentiable 113 §3.2. 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. 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- 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 '') 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. Naturally the themes discussed here are only a small choice from a great variety of possibilities. This choice has been shaped by the author's own predilections and by his work. Nevertheless the author hopes that his book presents a good foundation for the study of the more advanced literature indicated in the bibliography. I thank Sigrid Guttner and Robert Wetke most sincerely for their careful preparation of the majority of the ET^X files. Robert Wetke also deserves special thanks for the preparation of the computer diagrams. I am most grateful to Dr. Michael Lonne and Dr. Jorg Zintl for their help in proof• reading.

Hanover, January 2001 Wolfgang Ebeling List of figures

1.1 Transition function 1 1.2 Lattice L and parallelogram P 3 1.3 Definition of a holomorphic map 6 1.4 A homotopy F between 71 and 72 10 1.5 The homotopy FG 11 1.6 The homotopy F between the constant path XQ and 77"l 11 1.7 Analytic continuation along a path 27 1.8 The of the function i/A — z2 40

2.1 Polycylinder around 0 6 C2 46 2.2 Choice of the balls B\,...,Bt 75 2.3 The chart $ 85

3.1 Definition of a differentiate map 114 3.2 Tangent vector 115 3.3 Chart of a submanifold 118 3.4 Section of a differentiable fiber bundle 122 3.5 Tangent vector to a phase curve 124 3.6 Transversal - not transversal 126 3.7 Critical set C and discriminant D 153 3.8 AxTcS 156 3.9 The line C x {Xt} intersects the discriminant D transversally 157

3.10 X0 171

xiii XIV List of figures

3.11 Fibers of the map / 172

3.12 The level surface Xx 173 3.13 The path A 173

3.14 Xx(t) for t = 0,1/2,1 174 3.15 Graph of the bell function \ 174 3.16 Vanishing cycle S 175 3.17 Covanishing cycle £* 176

3.18 Image of 5 and 8* under ht 176 3.19 Effect of the monodromy h 176 3.20 The cycle <5* - h{6*) 176

3.21 The curve XR,0 for k = 6 177

3.22 The curve XR,0 for k = 7 178 3.23 The Coxeter-Dynkin diagram of type A^ 179 3.24 Coxeter-Dynkin diagrams of simple curve singularities 179

4.1 R% 181 4.2 Chart of a with boundary 182 4.3 Tangent space at a boundary point 183 4.4 Preferred orientation of the boundary 186 4.5 Construction of the vector field X 188 4.6 Vertical and horizontal tangent space 190 4.7 Parallel transport along the path 7 192 4.8 Standard 2-simplex 194 4.9 Example of a relative 1-cycle and of a relative 1-boundary 199 4.10 The excision theorem 200 4.11 Neighborhood U of A1 201 4.12 Orientation of A1 202 4.13 Example of (A, B) = 0 203 4.14 Proof of the Claim 204 4.15 Displacement of the zero section 207 4.16 The vector field X 208 4.17 Definition of the linking number 209 4.18 Another definition of the linking number 210 4.19 A braid with 3 strands 213 4.20 Plane projection of a braid 213 List of figures xv

4.21 The braid a, 213 4.22 A moroccan braid 214 4.23 Leather strip with slits 214

4.24 The unit ball I2 216 2 1 1 4.25 A map/: (I ,1 , J ) ^ (X, A, x0) 217 4.26 The homotopy H 218 4.27 The paths / and 7 219 4.28 The retraction of J« onto Jq~l 219 4.29 The cube I" x I 220

5.1 Vanishing cycle 227 5.2 Simple loop associated to 7 228 5.3 (Strongly) distinguished system of paths 228 5.4 The disc bundle DSn 232 5.5 The disc bundle DSn 234 5.6 The image of * 235 5.7 The (n + l)-cell e 238 5.8 Proof of Lemma 5.4 238

5.9 The vector field on X0 \ {0} 240 5.10 The homotopy g 241

5.11 The discs AVi 243 5.12 The sets V and W 244

5.13 The sets Xt and % 245 5.14 51V51V51VS1 247 5.15 The loop a; 248 5.16 u is homotopic to u^a^-i • • • UJ\ 251 5.17 Critical values of f\ 251 5.18 Riemann surface of y = ±(-z3 + 3A.?)1/2 251 5.19 Riemann surface of y = ±(—z3 + 3Xz + w)1/2 for «; = -2AA1/2,0,2AA1/2 252 X) 5.20 The fibers X^ for w = s1,0,s2 253 5.21 Coxeter-Dynkin diagram of type Ai 253 5.22 The map a 258 5.23 Retraction of c onto da 258 5.24 The operation aj 259 XVI List of figures

5.25 The operation Pj+1 260 5.26 The operation (1J+IQLJ 261 5.27 The operation ajaj+icxj 262 5.28 The operation aj+iajCtj+i 262 5.29 The braid b corresponding to a pair of strongly distinguished

path systems (71,72), (n, r2) 263 5.30 Critical values of the function f\ and the paths 7* and r 266 5.31 Coxeter-Dynkin diagram with respect to (5i,..., 5k) for k = 5 267 5.32 Coxeter-Dynkin diagram with respect to (5[,..., <%) for /c = 5 267 5.33 Coxeter-Dynkin diagram with respect to (5[ ,...... ,4fc~1)) for A:-5 268 5.34 Coxeter-Dynkin diagram with respect to (£1,..., £&) for fc = 5 268 5.35 Coxeter-Dynkin diagram of type A^ 268

5.36 Small discs around the points of Dt in St 271 5.37 New path 7 273 5.38 Definition of 7^ and T{ 273 5.39 The path 7 274 5.40 The map z ^ z2 Til 5.41 The local Milnor fibers % 278 5.42 Extension of the strongly distinguished path system (7i>--->7m) to (71,...,7M) 281 5.43 Choice of strongly distinguished path system (71,..., 7^) after Gabrielov 289 5.44 Coxeter-Dynkin diagram for the basis (5™) of the example 292 5.45 Coxeter-Dynkin diagram corresponding to 5' 296 5.46 Normal form of the Coxeter-Dynkin diagrams for the parabolic and hyperbolic unimodal singularities 298 5.47 Normal form of the Coxeter-Dynkin diagrams for the exceptional unimodal singularities 298 5.48 Coxeter-Dynkin diagram with respect to (5\,..., 5Q) 300 List of tables

5.1 The parabolic and hyperbolic unimodal singularities 293 5.2 The 14 exceptional unimodal singularities 294 5.3 Coxeter-Dynkin diagrams of the parabolic and hyperbolic unimodal singularities 295 5.4 Coxeter-Dynkin diagrams of the exceptional unimodal singularities 295

xvii

Bibliography

[A'C79] N. A'Campo: Tresses, monodromie et le groupe symplectique. Comment. Math. Helv. 54, 318-327 (1979). [Ahl78] L. V. Ahlfors: Complex Analysis. Third Edition. McGraw-Hill Book Co., New York, 1978. [Arn73] V. I. Arnold: Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities. Funct. Anal. Appl. 6, 254-272 (1973). [Arn06] V. I. Arnold: Ordinary Differential Equations. Springer-Verlag, Berlin, 2006. [AGV85] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko: Singularities of Differen- tiable Maps, I. Birkhauser-Verlag, Basel, 1985. [AGV88] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko: Singularities of Differen- tiable Maps, II. Birkhauser-Verlag, Basel, 1988. [Arn93] V. I. Arnold (Ed.): Dynamical Systems VI - Singularity Theory I. (Encyclopae• dia of Math. Sciences, Vol. 6) Springer-Verlag, New York, etc., 1993. [BK91] D. Battig, H. Knorrer: Singularitaten. Birkhauser-Verlag, Basel, 1991. [Bre93] G. Bredon: Geometry and Topology. (Graduate Texts in Math. 139) Springer- Verlag, New York, etc., 1993. [Bri70] E. Brieskorn: Die Monodromie der isolierten Singularitaten von Hyperflachen. Manuscripta Math. 2, 103-160 (1970). [Bri88] E. Brieskorn: Automorphic sets and braids and singularities. Contemporary Math. 78, 1988, pp. 45-115. [BK86] E. Brieskorn, H. Knorrer: Plane Algebraic Curves. Birkhauser-Verlag, Basel, 1986. [BJ82] Th. Brocker, K. Janich: Introduction to Differential Topology. Cambridge Uni• versity Press, Cambridge, 1982. [Chm82] S. V. Chmutov: Monodromy groups of critical points of functions. Invent. Math. 67, 123-131 (1982). [CJ89] R. Courant, F. John: Introduction to Calculus and Analysis. Volume I. Springer-Verlag, New York, etc., 1989.

303 304 Bibliography

[DFN85] B. A. Dubrovin, A. T. Fomenko, S. P. Novikov: Modern Geometry - Methods and Applications. Part II. The Geometry and Topology of Manifolds. (Graduate Texts in Math. 104) Springer-Verlag, New York, etc., 1985. [Dur79] A. H. Durfee: Fifteen characterizations of rational double points and simple critical points. Enseign. Math. 25, 131-163 (1979). [Ebe81] W. Ebeling: Quadratische Formen und Monodromiegruppen von Singu- laritaten. Math. Ann. 255, 463-498 (1981). [Ebe87] W. Ebeling: The Monodromy Groups of Isolated Singularities of Complete Intersections. (Lecture Notes in Math., Vol. 1293) Springer-Verlag, Berlin, etc., 1987. [Ebe96] W. Ebeling: On Coxeter-Dynkin diagrams of hypersurface singularities. J. of Math. Sciences 82, 3657-3664 (1996). [Fis76] G. Fischer: Complex Analytic Geometry. (Lecture Notes in Math., Vol. 538) Springer-Verlag, Berlin, etc., 1976. [FL88] W. Fischer, I. Lieb: Ausgewahlte Kapitel aus der Funktionentheorie. Vieweg, Braunschweig Wiesbaden, 1988. [FL92] W. Fischer, I. Lieb: Funktionentheorie, 6. Auflage. Vieweg, Braunschweig Wiesbaden, 1992. [For81] O. Forster: Lectures on Riemann Surfaces. (Graduate Texts in Math., Vol. 81) Springer-Verlag, New York, etc., 1981. [Gab79] A. M. Gabrielov: Polar curves and intersection matrices of singularities. In• vent. Math. 54, 15-22 (1979). [GG74] M. Golubitsky, V. Guillemin: Stable Mappings and their Singularities. (Grad• uate Texts in Math., Vol. 14) Springer-Verlag, Berlin, etc., 1974. [GF76] H. Grauert, K. Fritzsche: Several Complex Variables. (Graduate Texts in Math. 38) Springer-Verlag, New York, 1976. [GR71] H. Grauert, R. Remmert: Analytische Stellenalgebren. (Grundlehren der math. Wiss. 176) Springer-Verlag, Heidelberg, etc., 1971. [GR04] H. Grauert, R. Remmert: Theory of Stein Spaces. Springer-Verlag, Berlin, 2004. [GR84] H. Grauert, R. Remmert: Coherent Analytic Sheaves. (Grundlehren der math. Wiss. 265) Springer-Verlag, Heidelberg, etc., 1984. [GH81] M. J. Greenberg, J. R. Harper: Algebraic Topology: A First Course. Ben- jamin/Cummings, Menlo Park, CA, 1981. [GH78] P. Griffith, J. Harris: Principles of . Wiley and Sons, New York, 1978. [GR65] R. C. Gunning, H. Rossi: Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs, N. J., 1965. [Hir76] M. W. Hirsch: Differential Topology. (Graduate Texts in Math. 33) Springer- Verlag, Berlin, etc., 1976. [Hum85] S. P. Humphries: On weakly distinguished bases and free generating sets of free groups. Quart. J. Math. Oxford (2), 36, 215-219 (1985). [Ily87] G. G. H'yiita: On the Coxeter transformation of an isolated singularity. Russ. Math. Surveys 42:2, 279-280 (1987). [Jan83] W. A. M. Janssen: Skew-symmetric vanishing lattices and their monodromy groups. Math. Ann. 266, 115-133 (1983). Bibliography 305

[Jan85] W. A. M. Janssen: Skew-symmetric vanishing lattices and their monodromy groups II. Math. Ann. 272, 17-22 (1985). [JS87] G. A. Jones, D. Singerman: Complex Functions: An Algebraic and Geometric Viewpoint. Cambridge University Press, Cambridge, 1987. [dJPOO] Th. de Jong, G. Pfister: Local Analytic Geometry. Vieweg, Braunschweig Wies• baden, 2000. [KK83] L. Kaup, B. Kaup: Holomorphic Functions of Several Variables. Walter de Gruyter, Berlin, New York, 1983. [Kle73] F. Klein: Uber Flachen dritter Ordnung. Math. Ann. 6, 551-581 (1873) (Gesam- melte Math. Abhandlungen, Bd. II, J. Springer, Berlin, 1922, pp. 11-44, mit Zusatzen pp. 44-62). [Kod86] K. Kodaira: Complex Manifolds and Deformation of Complex Structures. (Grundlehren der math. Wiss. 283) Springer-Verlag, New York, etc., 1986. [Lam75] K. Lamotke: Die Homologie isolierter Singularitaten. Math. Z. 143, 27-44 (1975). [Lan97] S. Lang: Undergraduate Analysis. 2nd Edition. Springer-Verlag, New York, 1997. [Lan02] S. Lang: Algebra. 3rd Edition. (Graduate Texts in Math. 211) Springer-Verlag, New York, 2002. [Lef75] S. Lefschetz: Applications of Algebraic Topology. (Applied Math. Sciences, Vol. 16) Springer-Verlag, Berlin, etc., 1975. [Loj91] S. Lojasiewicz: Introduction to Complex Analytic Geometry. Birkhauser, Basel, 1991. [Loo84] E. J. N. Looijenga: Isolated Singular Points on Complete Intersections. (Lon• don Math. Soc. Lecture Note Series 77) Cambridge University Press, Cam• bridge, 1984. [LS77] R. C. Lyndon, P. E. Schupp: Combinatorial Group Theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete 89) Springer-Verlag, Berlin, Heidelberg, New York, 1977. [Mal68] B. Malgrange: Analytic spaces. In: Topics of Several Variables. L'Enseignement Math. Monographie 14, Geneve, 1968, pp. 1-28. [Mar82] J. Martinet: Singularities of Smooth Functions and Maps. (London Math. Soc. Lecture Note Series 58) Cambridge University Press, Cambridge, 1982. [Mil63] J. Milnor: Morse Theory. (Ann. of Math. Studies 51) Princeton University Press, Princeton, 1963. [Mil65] J. Milnor: Topology from the Differentiable Viewpoint. The University Press of Virginia, Charlottesville, 1965. [Mil68] J. Milnor: Singular Points of Complex Hypersurfaces. (Ann. of Math. Studies 61) Princeton University Press, Princeton, 1968. [MK71] J. Morrow, K. Kodaira: Complex Manifolds. Holt, Rinehart, and Winston, New York, 1971. [Mum76] D. Mumford: Algebraic Geometry I, Complex Projective Varieties. Springer- Verlag, Berlin, etc., 1976. [Nar66] R. Narasimhan: Introduction to the Theory of Analytic Spaces. (Lecture Notes in Math., Vol. 25) Springer-Verlag, Berlin, etc., 1966. [Nar85] R. Narasimhan: Analysis on Real and Complex Manifolds. Third printing. North-Holland, Amsterdam, New York, Oxford, 1985. 306 Bibliography

[Orl76] P. Orlik: The multiplicity of a holomorphic map at an isolated critical point. In: Real and Complex Singularities (P. Holm, ed.), Proc. Nordic Summer School, Oslo 1976, Sijthoff & Noordhoff, Alphen a/d Rijn 1976, pp. 405-474. [RK82] W. Rothstein, K. Kopfermann: Funktionentheorie mehrerer komplexer Veranderlicher. B. I.-Wissenschaftsverlag, Mannheim, 1982. [Ste51] N. Steenrod: The Topology of Fibre Bundles. Princeton University Press, Princeton, 1951. [SZ88] R. Stocker, H. Zieschang: Algebraische Topologie. Teubner Verlag, Stuttgart, 1988. [Voi80] E. Voigt: Marokkanische Zopfe: eine mathematische Bastelanleitung. Diplom- arbeit, Bonn, 1980. [War83] F. Warner: Foundations of Differentiable Manifolds and Lie Groups. (Graduate Texts in Math., Vol. 94) Springer-Verlag, New York, etc., 1983. [Waj80] B. Wajnryb: On the monodromy group of singularities. Math. Ann. 246, 141-154 (1980). [Wel80] R. O. Wells: Differential Analysis of Complex Manifolds. (Graduate Texts in Math. 65) Springer-Verlag, New York, etc., 1980. [Whi72] H. Whitney: Complex Analytic Varieties. Addison-Wesley, Reading, Massa• chusetts, 1972. [Wol64] J. Wolf: Differentiable fibre spaces and mappings compatible with Riemannian metrics. Michigan J. Math. 11, 65-70 (1964). Index

m-modal, 158 biholomorphic, 5, 134 m-modular, 158 bimodal, 158 r-determined, 159 bimodular, 158 r-, 159 boundary, 182, 195 relative, 198 A-sequence, 106 boundary operator, 217 Abel's lemma, 49 boundary point, 182 action, 133, 140 bouquet of n-spheres, 247 active, 104 braid, 212 active lemma, 104 braid group, 211 acts, 21 branched meromorphic continuation, 29 admissible, 31, 289 bundle chart, 119 algebra analytic, 81 Cauchy's integral formula, 46 local, 81 cell, 237 algebraic function, 33 chain analytic, 51 differentiable, 201 analytic algebra, 81 singular, 194 analytic continuation, 27 chart, 1, 182 complete, 30 Chevalley dimension, 102 analytic set germs, 76 classical geometric monodromy, 239 analytic subset, 74 classical monodromy, 239 annihilator ideal, 107 classical monodromy operator, 239 associated, 107 closed manifold, 182 atlas, 1 CM-ring, 108 complex, 2, 134 codimension, 76, 118, 139 holomorphic, 2, 134 Cohen-Macaulay ring, 108 attaching a cell, 237 complete, 29 augmentation, 196 complete analytic continuation, 30 automorphism, 275 complete intersection, 106 complex atlas, 2, 134 base point, 12 complex differentiable, 44 basis, 119 complex , 140 countable, 184 complex manifold, 134 symplectic, 300 complex Morse lemma, 142 Betti number, 197 complex structure, 2, 134, 137

307 308 Index

complex submanifold, 84, 139 differentiable partition of unity, 184 components differentiable path, 115 irreducible, 83 differentiable simplex, 200 conformally equivalent, 5 differentiable structure, 113 congruence subgroup, 300 differentiable submanifold, 118 connected differential, 118, 122, 137 simply, 12 dimension, 102, 134 semi-locally, 18 Chevalley, 102 connecting homomorphism, 199, 217 Weierstrass, 102 connection disc bundle, 184 Ehresmann, 191 discriminant, 35 continuation distinguished analytic, 27 strongly, 228, 249 complete, 30 weakly, 229, 249 meromorphic divides, 72 branched, 29 division theorem contractible, 193 special, 67 convergent, 48, 61 domain, 43 normally, 48 dual module, 298 coordinate local, 4 Ehresmann connection, 191 coordinate neighborhood, 4 Ehresmann fibration theorem, 186 cotangent space, 136 element countable basis, 184 integral, 99 covanishing cycle, 175 neutral, 215 covering, 13 embedding, 126 universal, 17 primitive, 282 covering transformation, 20 equation of the hypersurface, 84 Coxeter-Dynkin diagram, 179, 249 equivalent, 2, 18, 24, 60, 113, 134, 144 critical point, 127, 140 conformally, 5 isolated, 141 essential singularity, 5 critical value, 127, 140 Euler characteristic, 197 cross ratio, 41 even, 248 curve selection lemma, 141 excision theorem, 200 cycle, 195 exponential map, 129 covanishing, 175 monotone, 298 face, 195, 215 relative, 198 factorial, 72 vanishing, 175, 227, 249 fiber, 20, 119 fiber bundle deformation retract, 215 differentiable, 119 deforms to, 282 fiber preserving, 20 degenerate, 142 fibration theorem dense Ehresmann, 186 nowhere, 75 finite, 96, 99 derivation, 115, 135 fixed point, 21 , 44 fixed point free, 21 partial, 44 formal power series, 47 diagram fractional linear transformation, 41 Coxeter-Dynkin, 179, 249 function , 114 algebraic, 33 differentiable, 113, 114 meromorphic, 7 complex, 44 function germ, 24, 60 differentiable chain, 201 functional matrix differentiable fiber bundle, 119 holomorphic, 57 differentiable manifold, 113 fundamental domain, 3 differentiable manifold with boundary, 182 fundamental group, 12 Index 309

fundamental parallelogram, 3 infinitesimally versal, 146 integral, 99 Gabrielov numbers, 292 integral element, 99 generalized Morse lemma, 153 integral formula geometric monodromy, 175, 224 Cauchy's, 46 classical, 239 interior, 182 geometric monodromy group, 193 intersection complete, 106 Hensel's lemma, 73 intersection form, 205 Hilbert basis theorem, 70 intersection index, 202, 203 holomorphic, 4, 5, 45, 57, 89, 134 intersection matrix, 249 holomorphic atlas, 2, 134 irreducible, 71, 82, 112 holomorphic functional matrix, 57 irreducible components, 83 holomorphic local one parameter group, isolated critical point, 141 140 isolated singularity, 4, 141 holomorphic one parameter group, 140 isomorphic, 5 holomorphic vector field, 140 isomorphism, 96 homeomorphism isotopic, 192 local, 14 isotopy, 192 homologous, 196 isotropy group, 133 homology group, 196 reduced, 196 Jacobi matrix, 57 relative, 198 homology sequence of the pair, 199 Krull's intersection theorem, 93 homomorphism induced, 197, 216 lattice, 3, 248 homotopic, 9, 193, 260 vanishing, 276 homotopic relative to, 215 Lefschetz-Poincare duality, 208 homotopy, 9 left invariant, 128 homotopy equivalent, 246 lemma homotopy group, 215 active, 104 relative, 216 length, 106 homotopy lifting theorem, 193 Lie group, 127 homotopy sequence of the differentiable complex, 140 fiber bundle, 221 Lie subgroup, 130 homotopy sequence of the triple, 217 lifting, 14 horizontal tangent space, 190 lifting theorem, 94 hyperbolic, 297 linear map of vector bundles, 122 hyperbolic singularities, 297 link, 256 hypersurface, 84 linking number, 209 local, 90 ideal, 62 local algebra, 81 maximal, 62 local coordinates, 4 radical, 80 local homeomorphism, 14 ideal of a set germ, 77 local one parameter group of identification with a point, 209 , 123 identity theorem, 6 local ring, 62 identity theorem for holomorphic functions, locally analytic subset, 74 56 locally compact, 184 identity theorem for power series, 56 locally finite, 184 immersion, 126 locally path connected, 17 implicit function theorem, 59 loop index, 206 simple, 227 induced, 123 induced homomorphism, 197, 216 Malgrange preparation theorem, 70 induced unfolding, 145 manifold inertia indices, 293 closed, 182 310 Index

complex, 134 open, 120 different iable, 113 orbit, 21, 133 Riemannian, 183 order, 8, 64 topological, 1 orientable, 185 manifold with boundary orientation, 185 differentiate, 182 preferred, 185 topological, 181 orientation of the boundary map germ, 90 preferred, 186 maximal ideal, 62 meromorphic continuation pair, 198 branched, 29 parabolic singularities, 296 meromorphic function, 7 paracompact, 184 metric parallel transport, 192 Riemannian, 183 parameter Milnor fiber, 239 local uniformizing, 4 Milnor fibration, 239 parameter system, 102 Milnor lattice, 248 partial derivative, 44 , 150 partition of unity miniversal, 145 different iable, 184 modality, 158 subordinate to a covering, 184 module path, 9 dual, 298 different iable, 115 module number, 158 path connected, 12 monic polynomial, 73 locally, 17 monodromy, 173, 224 phase curve, 123 classical, 239 Picard-Lefschetz formula, 177 geometric, 175, 224 Picard-Lefschetz formulae, 236 classical, 239 Picard-Lefschetz theorem, 232 monodromy group, 226, 249, 276 Picard-Lefschetz transformation, 227 geometric, 193 Poincare-Hopf theorem, 208 monodromy operator, 224 point classical, 239 critical, 127, 140 monodromy theorem, 28 isolated, 141 monotone cycle, 298 regular, 85 Morse function, 144 singular, 85 Morse lemma pointing inwards, 183 complex, 142 pointing outwards, 183 generalized, 153 polar curve, 284 morsification, 149 pole, 5 real, 177 polycylinder, 46 multiplicity, 8, 150 polynomial monic, 73 Nakayama's lemma, 93 power series neutral element, 215 formal, 47 nilpotent, 81 preferred orientation, 185 nilradical, 81 preferred orientation of the boundary, 186 Noether normalization theorem, 101 preparation theorem for modules Noetherian, 70 Weierstrass, 97 nondegenerate, 142 prime, 72 normalization theorem prime sequence, 106 Noether, 101 primitive, 278 normally convergent, 48 primitive embedding, 282 nowhere dense, 75 product, 10 nowhere separating, 75 projection, 119 Puiseux series, 40 one parameter group of diffeomorphisms, 123 radical, 78 Index 311

radical ideal, 80 singularities rank, 125 hyperbolic, 297 rank theorem, 85, 125 parabolic, 296 real modification, 177 singularity, 140 reduced, 81 essential, 5 reduced homology group, 196 isolated, 4, 141 reducible, 71, 82, 112 removable, 5 refinement, 184 skew symmetric, 248 region, 43 smooth, 201 regular, 64, 105 special division theorem, 67 regular point, 85 spinor norm, 299 relative boundary, 198 stabilization, 290 relative cycle, 198 standard complex structure, 137 relative homology group, 198 standard simplex, 194 relative homotopy group, 216 star shaped, 13 removable singularity, 5 strongly distinguished, 228, 249 removable singularity theorem structure Riemann, 5 complex, 2, 134, 137 representative, 76 differentiable, 113 resultant, 34 standard complex, 137 Riemann surface, 2 sub-bundle, 190 Riemann surface of the algebraic function, subgroup 30 Lie, 130 Riemann's removable singularity theorem, 5 submanifold Riemannian manifold, 183 complex, 84, 139 Riemannian metric, 183 differentiable, 118 right equivalent, 158 , 126 ring subordinate to partition of unity, 184 CM-ring, 108 subset Cohen-Macaulay, 108 analytic, 74 local, 62 locally analytic, 74 ring of holomorphic function germs, 25, 61 summable, 91 root lattice, 277 support, 194 Riickert, basis theorem, 70 surface, 2 Riickert's Nullstellensatz, 78 Riemann, 2 symmetric, 248 Sard's theorem, 127 symplectic basis, 300 section, 122 section number, 203, 204 tangent bundle, 121 Seifert form, 257 tangent map, 118, 122, 137 semi-locally simply connected, 18 tangent space, 115, 136 separating horizontal, 190 nowhere, 75 vertical, 190 set germs Zariski, 136 analytic, 76 tangent vector, 115 sheaf of germs of holomorphic functions, 26 topological manifold, 1 sheets, 13 topological manifold with boundary, 181 shrinking to a point, 209 torus, 4 simple, 158 total space, 119 simple loop, 227 transformation simplex fractional linear, 41 differentiable, 200 Picard-Lefschetz, 227 singular, 194 transition function, 2 simply connected, 12 transversal, 126 semi-locally, 18 transversally, 201 singular chain, 194 triple, 215 singular simplex, 194 trivial, 119 312

tubular neighborhood, 256

unfolding, 144 induced, 145 unimodal, 158 unimodular, 158 unique factorization domain, 72 unit, 62 universal, 145 universal covering, 17 value, 24, 60 critical, 127, 140 vanishing cycle, 175, 227, 249 vanishing lattice, 276 variation, 225, 239 variation operator, 225 vector bundle, 120 vector field, 122 holomorphic, 140 versal, 145 infinitesimally, 146 vertical tangent space, 190 weakly distinguished, 229, 249 Weierstrass dimension, 102 Weierstrass division theorem, 65 Weierstrass polynomial, 65 Weierstrass preparation theorem, 65 Weierstrass preparation theorem for modules, 97 Weierstrass theorem, 52 Whitney sum, 191

Zariski tangent space, 136 zero section, 122 Titles in This Series

83 Wolfgang Ebeling, Functions of several complex variables and their singularities (translated by Philip G. Spain), 2007 82 Serge Alinhac and Patrick Gerard, Pseudo-differential operators and the Nash-Moser theorem (translated by Stephen S. Wilson), 2007 81 V. V. Prasolov, Elements of homology theory, 2007 80 Davar Khoshnevisan, Probability, 2007 79 William Stein, Modular forms, a computational approach (with an appendix by Paul E. Gunnells), 2007 78 Harry Dym, Linear algebra in action, 2007 77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton's Ricci flow, 2006 76 Michael E. Taylor, Measure theory and integration, 2006 75 Peter D. Miller, Applied asymptotic analysis, 2006 74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 72 R. J. Williams, Introduction the the mathematics of finance, 2006 71 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006 70 Sean Dineen, Probability theory in finance, 2005 69 Sebastian Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. Korner, A companion to analysis: A second first and first second course in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations II, 2003 59 Steven H. Weintraub, of finite groups: algebra and arithmetic, 2003 58 Cedric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003 55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003 54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and , 2002 51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002 50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002 49 John R. Harper, Secondary cohomology operations, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the /i-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 TITLES IN THIS SERIES

46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing

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