The Monodromy Group

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