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. It ends with applications to recent research questions related to mirror symmetry. The fundamental tool used is that of a vector bundle with connection. The book includes complete proofs, and applications to recent research questions. Aimed at graduate students and researchers, the book assumes some familiarity with basic complex algebraic geometry.
Author(s): Claude Sabbah
Series: Universitext
Edition: 1st Edition.
Publisher: Springer
Year: 2008
Language: English
Pages: 290
Cover......Page 1
Isomonodromic
Deformations and
Frobenius Manifolds:
An Introduction......Page 3
ISBN-13: 978-1-84800-053-7......Page 4
Preface......Page 8
Terminology and notation......Page 12
Holomorphic functions on an open set of Cn......Page 14
Complex analytic manifolds......Page 15
Holomorphic vector bundle......Page 18
Locally free sheaves of OM-modules......Page 20
Nonabelian cohomology......Page 23
Cech cohomology......Page 27
Line bundles......Page 29
Meromorphic bundles, lattices......Page 30
Examples of holomorphic and meromorphic bundles......Page 32
Affine varieties, analytization, algebraic differential forms......Page 38
Holomorphic connections on a vector bundle......Page 40
Holomorphic integrable connections and Higgs fields......Page 45
Geometry of the tangent bundle......Page 50
Meromorphic connections......Page 57
Locally constant sheaves......Page 61
Integrable deformations and isomonodromic deformations......Page 66
Appendix: the language of categories......Page 70
Cohomology of C, C* and P1......Page 73
Line bundles on P1......Page 75
A finiteness theorem and some consequences......Page 80
Structure of vector bundles on P1......Page 81
Families of vector bundles on P1......Page 88
Statement of the problems......Page 95
Local study of regular singularities......Page 97
Applications......Page 109
Complements......Page 112
Irregular singularities: local study......Page 114
The Riemann-Hilbert correspondence in the irregular case......Page 121
Lattices......Page 133
Lattices of (bold0mu mumu kk--@let@token -kkkk,)-vector spaces with regular singularity......Page 134
Lattices of (bold0mu mumu kk--@let@token -kkkk,)-vector spaces with an irregular singularity......Page 145
The Riemann-Hilbert problem and Birkhoff's problem......Page 156
The Riemann-Hilbert problem......Page 157
Meromorphic bundles with irreducible connection......Page 163
Application to the Riemann-Hilbert problem......Page 166
Complements on irreducibility......Page 169
Birkhoff's problem......Page 170
Fourier-Laplace duality......Page 178
Modules over the Weyl algebra......Page 179
Fourier transform......Page 187
Fourier transform and microlocalization......Page 194
Integrable deformations of bundles with connection on the Riemann sphere......Page 201
The Riemann-Hilbert problem in a family......Page 202
Birkhoff's problem in a family......Page 210
Universal integrable deformation for Birkhoff's problem......Page 218
Saito structures and Frobenius structures on a complex analytic manifold......Page 232
Saito structure on a manifold......Page 233
Frobenius structure on a manifold......Page 242
Infinitesimal period mapping......Page 246
Examples......Page 251
Frobenius-Saito structure associated to a singularity......Page 263
References......Page 271
Index of Notation......Page 280
Index......Page 282
