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The Monodromy Group

Henry Żołądek
Publisher: 
Birkhäuser
Publication Date: 
2006
Number of Pages: 
580
Format: 
Hardcover
Series: 
Monograpie Matematyczne 67
Price: 
129.00
ISBN: 
3-7643-7535-3
Category: 
Monograph
We do not plan to review this book.

Preface vii

1 Analytic Functions and Morse Theory 1

ァ1 TheoremaboutMonodromy . . . . . . . . . . . . . . . . . . . . . . 1

ァ2 Morse Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

ァ3 TheMorse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Normal Forms of Functions 13

ァ1 Tougeron Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

ァ2 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

ァ3 Proofs of Theorems 2.3 and 2.4 . . . . . . . . . . . . . . . . . . . . 23

ァ4 Classification of Singularities . . . . . . . . . . . . . . . . . . . . . 29

3 Algebraic Topology of Manifolds 35

ァ1 Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . 35

ァ2 Index of Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . 40

ァ3 Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Topology and Monodromy of Functions 57

ァ1 Topology of a Non-singular Level . . . . . . . . . . . . . . . . . . . 57

ァ2 Picard-Lefschetz Formula . . . . . . . . . . . . . . . . . . . . . . . 65

ァ3 Root Systems and CoxeterGroups . . . . . . . . . . . . . . . . . . 82

ァ4 BifurcationalDiagrams . . . . . . . . . . . . . . . . . . . . . . . . . 88

ァ5 Resolution and Normalization . . . . . . . . . . . . . . . . . . . . . 102

5 Integrals along Vanishing Cycles 117

ァ1 Analytic Properties of Integrals . . . . . . . . . . . . . . . . . . . . 117

ァ2 Singularities and Branching of Integrals . . . . . . . . . . . . . . . 125

ァ3 Picard–Fuchs Equations . . . . . . . . . . . . . . . . . . . . . . . . 128

ァ4 Gauss–Manin Connection . . . . . . . . . . . . . . . . . . . . . . . 140

ァ5 Oscillating Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6 Vector Fields and Abelian Integrals 159

ァ1 Phase Portraits of Vector Fields . . . . . . . . . . . . . . . . . . . . 159

ァ2 Method of Abelian Integrals . . . . . . . . . . . . . . . . . . . . . . 164

ァ3 Quadratic Centers and Bautin’s Theorem . . . . . . . . . . . . . . 189

vi Contents

7 Hodge Structures and Period Map 195

ァ1 Hodge Structure on AlgebraicManifolds . . . . . . . . . . . . . . . 196

ァ2 Hypercohomologies and Spectral Sequences . . . . . . . . . . . . . 203

ァ3 Mixed Hodge Structures . . . . . . . . . . . . . . . . . . . . . . . . 210

ァ4 Mixed Hodge Structures andMonodromy . . . . . . . . . . . . . . 224

ァ5 PeriodMapping in Algebraic Geometry . . . . . . . . . . . . . . . 252

8 Linear Differential Systems 267

ァ1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

ァ2 Regular Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 270

ァ3 Irregular Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 279

ァ4 Global Theory of Linear Equations . . . . . . . . . . . . . . . . . . 293

ァ5 Riemann–Hilbert Problem . . . . . . . . . . . . . . . . . . . . . . . 296

ァ6 The Bolibruch Example . . . . . . . . . . . . . . . . . . . . . . . . 307

ァ7 IsomonodromicDeformations . . . . . . . . . . . . . . . . . . . . . 315

ァ8 Relation with QuantumField Theory . . . . . . . . . . . . . . . . 324

9 Holomorphic Foliations. Local Theory 333

ァ1 Foliations and Complex Structures . . . . . . . . . . . . . . . . . . 334

ァ2 Resolution for Vector Fields . . . . . . . . . . . . . . . . . . . . . . 339

ァ3 One-DimensionalAnalytic Diffeomorphisms . . . . . . . . . . . . . 346

ァ4 The Ecalle Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 360

ァ5 Martinet–RamisModuli . . . . . . . . . . . . . . . . . . . . . . . . 366

ァ6 Normal Forms for Resonant Saddles . . . . . . . . . . . . . . . . . 378

ァ7 Theorems of Briuno and Yoccoz . . . . . . . . . . . . . . . . . . . . 381

10 Holomorphic Foliations. Global Aspects 393

ァ1 Algebraic Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

ァ2 Monodromy of the Leaf at Infinity . . . . . . . . . . . . . . . . . . 411

ァ3 Groups of Analytic Diffeomorphisms . . . . . . . . . . . . . . . . . 418

ァ4 The Ziglin Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

11 The Galois Theory 441

ァ1 Picard–Vessiot Extensions . . . . . . . . . . . . . . . . . . . . . . . 441

ァ2 TopologicalGalois Theory . . . . . . . . . . . . . . . . . . . . . . . 471

12 Hypergeometric Functions 491

ァ1 The Gauss Hypergeometric Equation . . . . . . . . . . . . . . . . . 491

ァ2 The Picard–Deligne–MostowTheory . . . . . . . . . . . . . . . . . 515

ァ3 Multiple Hypergeometric Integrals . . . . . . . . . . . . . . . . . . 527

Bibliography 537

Index 559

Dummy View - NOT TO BE DELETED