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 t h e Ge rman language by Friedr. Vieweg & Sohn Verlag, D-65189 Wiesbaden, Germany,

as "Wolfgang Ebeling: Funkt ionentheor ie , Differentialtopologie und Singular i ta ten. 1. Auflage ( 1 s t ed i t ion)" .

© Friedr. Vieweg & Sohn Verlag | G W V Fachverlage G m b H , Wiesbaden , 2001

Transla ted by Phi l ip G. Spain

2000 Mathematics Subject Classification. P r i m a r y 32 -01 ; Secondary 32S10, 32S55, 58K40, 58K60.

For addi t ional information and upda te s on this book, visit w w w . a m s . o r g / b o o k p a g e s / g s m - 8 3

Library of Congress Cataloging-in-Publicat ion D a t a

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

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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 improve­ments 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 alge­braic/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 par­ticular 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 func­tions of several complex variables and on this basis to develop the fundamen­tal concepts of the theory of isolated singularities of holomorphic functions systematically. It is derived from lectures given by the author to mathe­matics 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 intro­ductory 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 func­tions of several complex variables. They also present an introduction to local complex geometry. In the third chapter the results will be applied to defor­mation 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

Xl l 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 singulari­ties 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 ex­tensive 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 vari­ables, 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 vari­ables. 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 in­dependently 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 proof­reading.

Hanover, January 2001 Wolfgang Ebeling

List of figures

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

2.1

2.2

2.3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

Transition function

Lattice L and parallelogram P

Definition of a holomorphic map

A homotopy F between 71 and 72

The homotopy FG

The homotopy F between the constant path XQ and 7 7 " l

Analytic continuation along a path

The Riemann surface of the function i/A — z2

Polycylinder around 0 6 C2

Choice of the balls B\,...,Bt

The chart $

Definition of a differentiate map

Tangent vector

Chart of a submanifold

Section of a differentiable fiber bundle

Tangent vector to a phase curve

Transversal - not transversal

Critical set C and discriminant D

A x T c S The line C x {Xt} intersects the discriminant D transversally

X0

1

3

6

10

11

11

27

40

46

75

85

114

115

118

122

124

126

153

156

157

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 manifold 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

4.25 A m a p / : (I2,11, J1) ^ (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 5 1 V 5 1 V 5 1 V S 1 247

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

5.20 The fibers X^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 (5 i , . . . , 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[ , . . .

. . . ,4 f c~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 ( 7 1 , . . . , 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

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Index

m-modal, 158 m-modular, 158 r-determined, 159 r-jet, 159

A-sequence, 106 Abel's lemma, 49 action, 133, 140 active, 104 active lemma, 104 acts, 21 admissible, 31, 289 algebra

analytic, 81 local, 81

algebraic function, 33 analytic, 51 analytic algebra, 81 analytic continuation, 27

complete, 30 analytic set germs, 76 analytic subset, 74 annihilator ideal, 107 associated, 107 atlas, 1

complex, 2, 134 holomorphic, 2, 134

attaching a cell, 237 augmentation, 196 automorphism, 275

base point, 12 basis, 119

countable, 184 symplectic, 300

Betti number, 197

biholomorphic, 5, 134 bimodal, 158 bimodular, 158 boundary, 182, 195

relative, 198 boundary operator, 217 boundary point, 182 bouquet of n-spheres, 247 braid, 212 braid group, 211 branched meromorphic continuation, 29 bundle chart, 119

Cauchy's integral formula, 46 cell, 237 chain

differentiable, 201 singular, 194

chart, 1, 182 Chevalley dimension, 102 classical geometric monodromy, 239 classical monodromy, 239 classical monodromy operator, 239 closed manifold, 182 CM-ring, 108 codimension, 76, 118, 139 Cohen-Macaulay ring, 108 complete, 29 complete analytic continuation, 30 complete intersection, 106 complex atlas, 2, 134 complex differentiable, 44 complex Lie group, 140 complex manifold, 134 complex Morse lemma, 142 complex structure, 2, 134, 137

307

308 Index

complex submanifold, 84, 139 components

irreducible, 83 conformally equivalent, 5 congruence subgroup, 300 connected

simply, 12 semi-locally, 18

connecting homomorphism, 199, 217 connection

Ehresmann, 191 continuation

analytic, 27 complete, 30

meromorphic branched, 29

contractible, 193 convergent, 48, 61

normally, 48 coordinate

local, 4 coordinate neighborhood, 4 cotangent space, 136 countable basis, 184 covanishing cycle, 175 covering, 13

universal, 17 covering transformation, 20 Coxeter-Dynkin diagram, 179, 249 critical point, 127, 140

isolated, 141 critical value, 127, 140 cross ratio, 41 curve selection lemma, 141 cycle, 195

covanishing, 175 monotone, 298 relative, 198 vanishing, 175, 227, 249

deformation retract, 215 deforms to, 282 degenerate, 142 dense

nowhere, 75 derivation, 115, 135 derivative, 44

partial, 44 diagram

Coxeter-Dynkin, 179, 249 diffeomorphism, 114 differentiable, 113, 114

complex, 44 differentiable chain, 201 differentiable fiber bundle, 119 differentiable manifold, 113 differentiable manifold with boundary, 182

differentiable partition of unity, 184 differentiable path, 115 differentiable simplex, 200 differentiable structure, 113 differentiable submanifold, 118 differential, 118, 122, 137 dimension, 102, 134

Chevalley, 102 Weierstrass, 102

disc bundle, 184 discriminant, 35 distinguished

strongly, 228, 249 weakly, 229, 249

divides, 72 division theorem

special, 67 domain, 43 dual module, 298

Ehresmann connection, 191 Ehresmann fibration theorem, 186 element

integral, 99 neutral, 215

embedding, 126 primitive, 282

equation of the hypersurface, 84 equivalent, 2, 18, 24, 60, 113, 134, 144

conformally, 5 essential singularity, 5 Euler characteristic, 197 even, 248 excision theorem, 200 exponential map, 129

face, 195, 215 factorial, 72 fiber, 20, 119 fiber bundle

differentiable, 119 fiber preserving, 20 fibration theorem

Ehresmann, 186 finite, 96, 99 fixed point, 21 fixed point free, 21 formal power series, 47 fractional linear transformation, 41 function

algebraic, 33 meromorphic, 7

function germ, 24, 60 functional matrix

holomorphic, 57 fundamental domain, 3 fundamental group, 12

Index

fundamental parallelogram, 3

Gabrielov numbers, 292 generalized Morse lemma, 153 geometric monodromy, 175, 224

classical, 239 geometric monodromy group, 193

Hensel's lemma, 73 Hilbert basis theorem, 70 holomorphic, 4, 5, 45, 57, 89, 134 holomorphic atlas, 2, 134 holomorphic functional matrix, 57 holomorphic local one parameter group,

140 holomorphic one parameter group, 140 holomorphic vector field, 140 homeomorphism

local, 14 homologous, 196 homology group, 196

reduced, 196 relative, 198

homology sequence of the pair, 199 homomorphism

induced, 197, 216 homotopic, 9, 193, 260 homotopic relative to, 215 homotopy, 9 homotopy equivalent, 246 homotopy group, 215

relative, 216 homotopy lifting theorem, 193 homotopy sequence of the differentiable

fiber bundle, 221 homotopy sequence of the triple, 217 horizontal tangent space, 190 hyperbolic, 297 hyperbolic singularities, 297 hypersurface, 84

ideal, 62 maximal, 62 radical, 80

ideal of a set germ, 77 identification with a point, 209 identity theorem, 6 identity theorem for holomorphic functions,

56 identity theorem for power series, 56 immersion, 126 implicit function theorem, 59 index, 206 induced, 123 induced homomorphism, 197, 216 induced unfolding, 145 inertia indices, 293

309

infinitesimally versal, 146 integral, 99 integral element, 99 integral formula

Cauchy's, 46 interior, 182 intersection

complete, 106 intersection form, 205 intersection index, 202, 203 intersection matrix, 249 irreducible, 71, 82, 112 irreducible components, 83 isolated critical point, 141 isolated singularity, 4, 141 isomorphic, 5 isomorphism, 96 isotopic, 192 isotopy, 192 isotropy group, 133

Jacobi matrix, 57

Krull's intersection theorem, 93

lattice, 3, 248 vanishing, 276

Lefschetz-Poincare duality, 208 left invariant, 128 lemma

active, 104 length, 106 Lie group, 127

complex, 140 Lie subgroup, 130 lifting, 14 lifting theorem, 94 linear map of vector bundles, 122 link, 256 linking number, 209 local, 90 local algebra, 81 local coordinates, 4 local homeomorphism, 14 local one parameter group of

diffeomorphisms, 123 local ring, 62 locally analytic subset, 74 locally compact, 184 locally finite, 184 locally path connected, 17 loop

simple, 227

Malgrange preparation theorem, 70 manifold

closed, 182

310 Index

complex, 134 different iable, 113 Riemannian, 183 topological, 1

manifold with boundary differentiate, 182 topological, 181

map germ, 90 maximal ideal, 62 meromorphic continuation

branched, 29 meromorphic function, 7 metric

Riemannian, 183 Milnor fiber, 239 Milnor fibration, 239 Milnor lattice, 248 Milnor number, 150 miniversal, 145 modality, 158 module

dual, 298 module number, 158 monic polynomial, 73 monodromy, 173, 224

classical, 239 geometric, 175, 224

classical, 239 monodromy group, 226, 249, 276

geometric, 193 monodromy operator, 224

classical, 239 monodromy theorem, 28 monotone cycle, 298 Morse function, 144 Morse lemma

complex, 142 generalized, 153

morsification, 149 real, 177

multiplicity, 8, 150

Nakayama's lemma, 93 neutral element, 215 nilpotent, 81 nilradical, 81 Noether normalization theorem, 101 Noetherian, 70 nondegenerate, 142 normalization theorem

Noether, 101 normally convergent, 48 nowhere dense, 75 nowhere separating, 75

one parameter group of diffeomorphisms, 123

open, 120 orbit, 21, 133 order, 8, 64 orientable, 185 orientation, 185

preferred, 185 orientation of the boundary

preferred, 186

pair, 198 parabolic singularities, 296 paracompact, 184 parallel transport, 192 parameter

local uniformizing, 4 parameter system, 102 partial derivative, 44 partition of unity

different iable, 184 subordinate to a covering, 184

path, 9 different iable, 115

path connected, 12 locally, 17

phase curve, 123 Picard-Lefschetz formula, 177 Picard-Lefschetz formulae, 236 Picard-Lefschetz theorem, 232 Picard-Lefschetz transformation, 227 Poincare-Hopf theorem, 208 point

critical, 127, 140 isolated, 141

regular, 85 singular, 85

pointing inwards, 183 pointing outwards, 183 polar curve, 284 pole, 5 polycylinder, 46 polynomial

monic, 73 power series

formal, 47 preferred orientation, 185 preferred orientation of the boundary, 186 preparation theorem for modules

Weierstrass, 97 prime, 72 prime sequence, 106 primitive, 278 primitive embedding, 282 product, 10 projection, 119 Puiseux series, 40

radical, 78

Index 311

radical ideal, 80 rank, 125 rank theorem, 85, 125 real modification, 177 reduced, 81 reduced homology group, 196 reducible, 71, 82, 112 refinement, 184 region, 43 regular, 64, 105 regular point, 85 relative boundary, 198 relative cycle, 198 relative homology group, 198 relative homotopy group, 216 removable singularity, 5 removable singularity theorem

Riemann, 5 representative, 76 resultant, 34 Riemann surface, 2 Riemann surface of the algebraic function,

30 Riemann's removable singularity theorem, 5 Riemannian manifold, 183 Riemannian metric, 183 right equivalent, 158 ring

CM-ring, 108 Cohen-Macaulay, 108 local, 62

ring of holomorphic function germs, 25, 61 root lattice, 277 Riickert, basis theorem, 70 Riickert's Nullstellensatz, 78

Sard's theorem, 127 section, 122 section number, 203, 204 Seifert form, 257 semi-locally simply connected, 18 separating

nowhere, 75 set germs

analytic, 76 sheaf of germs of holomorphic functions, 26 sheets, 13 shrinking to a point, 209 simple, 158 simple loop, 227 simplex

differentiable, 200 singular, 194

simply connected, 12 semi-locally, 18

singular chain, 194 singular simplex, 194

singularities hyperbolic, 297 parabolic, 296

singularity, 140 essential, 5 isolated, 4, 141 removable, 5

skew symmetric, 248 smooth, 201 special division theorem, 67 spinor norm, 299 stabilization, 290 standard complex structure, 137 standard simplex, 194 star shaped, 13 strongly distinguished, 228, 249 structure

complex, 2, 134, 137 differentiable, 113 standard complex, 137

sub-bundle, 190 subgroup

Lie, 130 submanifold

complex, 84, 139 differentiable, 118

submersion, 126 subordinate to partition of unity, 184 subset

analytic, 74 locally analytic, 74

summable, 91 support, 194 surface, 2

Riemann, 2 symmetric, 248 symplectic basis, 300

tangent bundle, 121 tangent map, 118, 122, 137 tangent space, 115, 136

horizontal, 190 vertical, 190 Zariski, 136

tangent vector, 115 topological manifold, 1 topological manifold with boundary, 181 torus, 4 total space, 119 transformation

fractional linear, 41 Picard-Lefschetz, 227

transition function, 2 transversal, 126 transversally, 201 triple, 215 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

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80 Davar Khoshnevisan, Probability, 2007

79 Wil l iam Stein, Modular forms, a computational approach (with an appendix by Paul E.

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78 Harry D y m , Linear algebra in action, 2007

77 Bennet t Chow, P e n g 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. Wil l iams, 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 Mi lman, and Antonis Tsolomit is , Functional analysis, An

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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, Representation theory of finite groups: algebra and arithmetic,

2003

58 Cedric Villani, Topics in optimal transportation, 2003

57 Robert P lato , 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 T h o m a s Friedrich, Global analysis: Differential forms in analysis,

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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

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45 Inder K. Rana, An introduction to measure and integration, second edition, 2002

44 J im Agler and John E. M c C a r t h y , Pick interpolation and Hilbert function spaces, 2002

43 N . V. Krylov, Introduction to the theory of random processes, 2002

42 J in 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,

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39 Larry C. Grove, Classical groups and geometric algebra, 2002

38 El ton P. Hsu, Stochastic analysis on manifolds, 2002

37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular

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36 Mart in 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 Dmitr i Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001

32 Robert G. Bart le , A modern theory of integration, 2001

31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods

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30 J. C. McConnel l 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 A lber to Candel and Lawrence Conlon, Foliations I, 2000

22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov

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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 Mart in Weese , Discovering modern set theory II: Set-theoretic

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17 Henryk Iwaniec, Topics in classical automorphic forms, 1997

16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator

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15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator

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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

For a complete list of t i t les in this series, visit t h e AMS Bookstore at www.ams.org/bookstore / .