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
