Isomonodromic Deformations and Frobenius Manifolds: An Introduction

Sabbah C.978-2-7598-0047-6

The notion of a Frobenius structure on a complex analytic manifold appeared at the end of the seventies in the theory of singularities of holomorphic functions. Motivated by physical considerations, further development of the theory has opened new perspectives on, and revealed new links between, many apparently unrelated areas of mathematics and physics.Based on a series of graduate lectures, this book provides an introduction to algebraic geometric methods in the theory of complex linear differential equations. Starting from basic notions in complex algebraic geometry, it develops some of the classical problems of linear differential equations and ends with applications to recent research questions related to mirror symmetry.The fundamental tool used within the book is that of a vector bundle with connection. There is a detailed analysis of the singularities of such objects and of their deformations, and coverage of the techniques used in the resolution of the Riemann-Hilbert problem and Birkhoff’s problem. An approach to Frobenius manifolds using isomonodromic deformations of linear differential equations is also developed.

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Table of contents :
Preface……Page 7
Terminology and notation……Page 11
Holomorphic functions on an open set of Cn……Page 13
Complex analytic manifolds……Page 14
Holomorphic vector bundle……Page 17
Locally free sheaves of OM-modules……Page 19
Nonabelian cohomology……Page 22
Cech cohomology……Page 26
Line bundles……Page 28
Meromorphic bundles, lattices……Page 29
Examples of holomorphic and meromorphic bundles……Page 31
Affine varieties, analytization, algebraic differential forms……Page 37
Holomorphic connections on a vector bundle……Page 39
Holomorphic integrable connections and Higgs fields……Page 44
Geometry of the tangent bundle……Page 49
Meromorphic connections……Page 56
Locally constant sheaves……Page 60
Integrable deformations and isomonodromic deformations……Page 65
Appendix: the language of categories……Page 69
Cohomology of C, C* and P1……Page 72
Line bundles on P1……Page 74
A finiteness theorem and some consequences……Page 79
Structure of vector bundles on P1……Page 80
Families of vector bundles on P1……Page 87
Statement of the problems……Page 94
Local study of regular singularities……Page 96
Applications……Page 108
Complements……Page 111
Irregular singularities: local study……Page 113
The Riemann-Hilbert correspondence in the irregular case……Page 120
Lattices……Page 132
Lattices of (bold0mu mumu kk–@let@token -kkkk,)-vector spaces with regular singularity……Page 133
Lattices of (bold0mu mumu kk–@let@token -kkkk,)-vector spaces with an irregular singularity……Page 144
The Riemann-Hilbert problem and Birkhoff’s problem……Page 155
The Riemann-Hilbert problem……Page 156
Meromorphic bundles with irreducible connection……Page 162
Application to the Riemann-Hilbert problem……Page 165
Complements on irreducibility……Page 168
Birkhoff’s problem……Page 169
Fourier-Laplace duality……Page 177
Modules over the Weyl algebra……Page 178
Fourier transform……Page 186
Fourier transform and microlocalization……Page 193
Integrable deformations of bundles with connection on the Riemann sphere……Page 200
The Riemann-Hilbert problem in a family……Page 201
Birkhoff’s problem in a family……Page 209
Universal integrable deformation for Birkhoff’s problem……Page 217
Saito structures and Frobenius structures on a complex analytic manifold……Page 231
Saito structure on a manifold……Page 232
Frobenius structure on a manifold……Page 241
Infinitesimal period mapping……Page 245
Examples……Page 250
Frobenius-Saito structure associated to a singularity……Page 262
References……Page 270
Index of Notation……Page 279
Index……Page 281