Table of contents for Isomonodromic deformations and Frobenius manifolds : an introduction / Claude Sabbah.


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Preface ......................................................  IX
Terminology and notation ................     .................. XIII
0     The language of fibre bundles ............................ 1
1   Holomorphic functions on an open set of C" ...... . ...... 1
2    Complex  analytic manifolds .............................  2
3    Holomorphic vector bundle .............................  5
4   Locally free sheaves of cAi-modules ...................... 7
5   Nonabelian cohomology ................................ 10
6    Cech cohomology .............................. ...... 14
7   Line bundles .......... ..  ..  ................  ....  .16
8   Meromorphic bundles, lattices ........................... 17
9   Examples of holomorphic and meromorphic bundles ....... 19
10  Affine varieties, analytization, algebraic differential forms . . 25
11  Holomorphic connections on a vector bundle .............. 27
12  Holomorphic integrable connections and Higgs fields ....... .32
13  Geometry of the tangent bundle .........................  37
14  Meromorphic connections ..........................  ....  44
15  Locally constant sheaves ............................... 48
16  Integrable deformations and isomonodromic deformations... 53
17  Appendix: the language of categories ................... . 57
I     Holomorphic vector bundles on the Riemann sphere ..... 61
1   Cohomology of C, C* and pI ............................ 61
2   Line bundles on P1 .................................... 63
3   A finiteness theorem and some consequences .............. 68
4   Structure of vector bundles on pl ........................ 69
5   Families of vector bundles on P! ......................... 76
II    The Riemann-Hilbert correspondence on a Riemann
surface  ..................................................  83
1   Statement of the problems ..............................  83
2    Local study of regular singularities .......................  85
3    Applications .......................................... 97
4    Complements ......................................... 100
5    Irregular singularities: local study  ....................... 102
6    The Riemann-Hilbert correspondence in the irregular case .. 109
III   Lattices .................................................121
1    Lattices of (k, V)-vector spaces with regular singularity .... 122
2    Lattices of (k, V)-vector spaces with an irregular singularity 133
IV    The Riemann-Hilbert problem and Birkhoff's problem       .. 145
1   The Riemann-Hilbert problem  .......................... 146
2    Meromorphic bundles with irreducible connection .......... 152
3    Application to the Riemann-Hilbert problem .............. 155
4    Complements on  irreducibility  .......................... 158
5    Birkhoff's problem .................................... 159
V     Fourier-Laplace  duality  .................................. 167
1   Modules over the Weyl algebra .......................... 168
2    Fourier transform  .....................................  176
3    Fourier transform  and  microlocalization  .................. 183
VI    Integrable deformations of bundles with connection on
the Riemann sphere .................................... 191
1   The Riemann-lHilbert problem in a family ................ 192
2    Birkhoff's problem  in  a  family ........................... 200
3    Universal integrable deformation for Birkhoff's problem .... 208
VII   Saito structures and Frobenius structures on a complex
analytic manifold ........................................ 223
1   Saito  structure on  a manifold  ........................... 224
2    Frobenius structure on  a manifold  ....................... 233
3    Infinitesimal period  mapping ............................ 237
4    Examples .................................... ... .........  242
5    Frobenius-Saito structure associated to a singularity ....... 254
R eferences ............................. ........................ 263
Index  of  N otation  ............................................. 273
Index .......................................................... 275



Library of Congress subject headings for this publication: Holomorphic functions, Vector bundles, Functions of several complex variables, Isomonodromic deformation method, Frobenius manifolds