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Mixed Hodge Structures and Singularities (Cambridge Tracts in Mathematics): Valentine S. Kulikov

This vital work is both an introduction to, and a survey of singularity theory, in particular, studying singularities by means of differential forms. Here, some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and the theory of period maps, are developed in the local situation to study the case of isolated singularities of holomorphic functions. The author introduces the Gauss-Manin connection on the vanishing cohomology of a singularity, that is on the cohomology fibration associated to the Milnor fibration, and draws on the work of Brieskorn and Steenbrink to calculate this connection, and the limit mixed Hodge structure. This is an excellent resource for all researchers in singularity theory, algebraic or differential geometry.

Table of Contents

Introduction

I The Gauss-Manin connection

1 Milnor fibration, Picard-Lefschetz monodromy transformation, topological Gauss-Manin connection 1

2 Connections, locally constant sheaves and systems of linear differential equations 3

3 De Rham cohomology 10

4 Gauss-Manin connection on relative De Rham cohomology 14

5 Brieskorn lattices 23

6 Absence of torsion in sheaves [actual symbol not reproducible] of isolated singularities 30

7 Singular points of systems of linear differential equations 33

8 Regularity of the Gauss-Manin connection 42

9 The monodromy theorem 46

10 Gauss-Manin connection of a non-isolated hypersurface singularity 51

II Limit mixed Hodge structure on the vanishing cohomology of an isolated hypersurface singularity

1 Mixed Hodge structures. Definitions. Deligne's theorem 60

2 The limit MHS according to Schmid 62

3 The limit MHS according to Steenbrink 73

4 Hodge theory of a smooth hypersurface according to Griffiths-Deligne 82

5 The Gauss-Manin system of an isolated singularity 88

6 Decomposition of a meromorphic connection into a direct sum of the root subspaces of the operator [actual symbol not reproducible]. The V-filtration and the canonical lattice 95

7 The limit Hodge filtration according to Varchenko and to Scherk-Steenbrink 103

8 Spectrum of a hypersurface singularity 115

III The period map of a -const deformation of an isolated hypersurface singularity associated with Brieskorn lattices and MHSs

1 Gluing of Milnor fibrations and meromorphic connections of a -const deformation of a singularity 139

2 Differentiation of geometric sections and their root components wrt a parameter 144

3 The period map 151

4 The infinitesimal Torelli theorem 165

5 The Picard-Fuchs singularity and Hertling's invariants 172

References 181

Index 185

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