Zoladek The Monodromy Group

1. Auflage 2006
ISBN: 978-3-7643-7536-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 67, 589 Seiten

Reihe: Monografie Matematyczne

In singularity theory and algebraic geometry, the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16 Hilbert problem and in mixed Hodge structures. There is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. In covering these and other topics, this book underlines the unifying role of the monogropy group.

Autoren/Hrsg.

Zoladek, Henryk

Inhaltsverzeichnis

1;Contents;5
2;Preface;7
3;1 Analytic Functions and Morse Theory;12
3.1;§1 Theorem about Monodromy;12
3.2;§2 Morse Lemma;14
3.3;§3 The Morse Theory;18
4;2 Normal Forms of Functions;23
4.1;§1 Tougeron Theorem;23
4.2;§2 Deformations;27
4.3;§3 Proofs of Theorems 2.3 and 2.4;33
4.4;§4 Classi.cation of Singularities;39
5;3 Algebraic Topology of Manifolds;45
5.1;§1 Homology and Cohomology;45
5.2;§2 Index of Intersection;50
5.3;§3 Homotopy Theory;65
6;4 Topology and Monodromy of Functions;67
6.1;§1 Topology of a Non-singular Level;67
6.2;§2 Picard-Lefschetz Formula;75
6.3;§3 Root Systems and Coxeter Groups;92
6.4;§4 Bifurcational Diagrams;98
6.5;§5 Resolution and Normalization;112
7;5 Integrals along Vanishing Cycles;127
7.1;§1 Analytic Properties of Integrals;127
7.2;§2 Singularities and Branching of Integrals;135
7.3;§3 Picard–Fuchs Equations;138
7.4;§4 Gauss–Manin Connection;150
7.5;§5 Oscillating Integrals;160
8;6 Vector Fields and Abelian Integrals;169
8.1;§1 Phase Portraits of Vector Fields;169
8.2;§2 Method of Abelian Integrals;174
8.3;§3 Quadratic Centers and Bautin’s Theorem;199
9;7 Hodge Structures and Period Map;205
9.1;§1 Hodge Structure on Algebraic Manifolds;206
9.2;§2 Hypercohomologies and Spectral Sequences;213
9.3;§3 Mixed Hodge Structures;220
9.4;§4 Mixed Hodge Structures and Monodromy;234
9.5;§5 Period Mapping in Algebraic Geometry;262
10;8 Linear Di.erential Systems;277
10.1;§1 Introduction;277
10.2;§2 Regular Singularities;280
10.3;§3 Irregular Singularities;289
10.4;§4 Global Theory of Linear Equations;303
10.5;§5 Riemann–Hilbert Problem;306
10.6;§6 The Bolibruch Example;317
10.7;§7 Isomonodromic Deformations;325
10.8;§8 Relation with Quantum Field Theory;334
11;9 Holomorphic Foliations. Local Theory;342
11.1;§1 Foliations and Complex Structures;343
11.2;§2 Resolution for Vector Fields;348
11.3;§3 One-Dimensional Analytic Di.eomorphisms;355
11.4;§4 The Ecalle Approach;369
11.5;§5 Martinet–Ramis Moduli;375
11.6;§6 Normal Forms for Resonant Saddles;387
11.7;§7 Theorems of Briuno and Yoccoz;390
12;10 Holomorphic Foliations. Global Aspects;401
12.1;§1 Algebraic Leaves;401
12.2;§2 Monodromy of the Leaf at In.nity;419
12.3;§3 Groups of Analytic Di.eomorphisms;426
12.4;§4 The Ziglin Theory;443
13;11 The Galois Theory;449
13.1;§1 Picard–Vessiot Extensions;449
13.2;§2 Topological Galois Theory;479
14;12 Hypergeometric Functions;499
14.1;§1 The Gauss Hypergeometric Equation;499
14.2;§2 The Picard–Deligne–Mostow Theory;523
14.3;§3 Multiple Hypergeometric Integrals;535
15;Bibliography;545
16;Index;566