Mixed Hodge structures

Chris A.M. Peters, Joseph H. M. Steenbrink3540770151, 9783540770152

This is the first comprehensive basic monograph on mixed Hodge structures. Starting with a summary of classic Hodge theory from a modern vantage point the book goes on to explain Deligne’s mixed Hodge theory. Here proofs are given using cubical schemes rather than simplicial schemes. Next come Hain’s and Morgan’s results on mixed Hodge structures related to homotopy theory. Steenbrink’s approach of the limit mixed Hodge structure is then explained using the language of nearby and vanishing cycle functors bridging the passage to Saito’s theory of mixed Hodge modules which is the subject of the last chapter. Since here D-modules are essential, these are briefly introduced in a previous chapter. At various stages applications are given, ranging from the Hodge conjecture to singularities. The book ends with three large appendices, each one in itself a resourceful summary of tools and results not easily found in one place in the existing literature (homological algebra, algebraic and differential topology, stratified spaces and singularities). The book is intended for advanced graduate students, researchers in complex algebraic geometry as well as interested researchers in nearby fields (algebraic geometry, mathematical physics

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Table of contents :
Cover……Page 1
A Series of Modern Surveys in Mathematics Volume 52……Page 2
Mixed Hodge Structures……Page 3
Preface……Page 5
Contents……Page 6
Introduction……Page 13
Part I Basic Hodge Theory……Page 21
1 Compact Kahler Manifolds……Page 22
2 Pure Hodge Structures……Page 44
3 Abstract Aspects of Mixed Hodge Structures……Page 72
Part II Mixed Hodge structures on Cohomology Groups……Page 97
4 Smooth Varieties……Page 98
5 Singular Varieties……Page 118
6 Singular Varieties: Complementary Results……Page 149
7 Applications to Algebraic Cycles and to Singularities……Page 168
Part III Mixed Hodge Structures on Homotopy Groups……Page 195
8 Hodge Theory and Iterated Integrals……Page 196
9 Hodge Theory and Minimal Models……Page 223
Part IV Hodge Structures and Local Systems……Page 241
10 Variations of Hodge Structure……Page 242
11 Degenerations of Hodge Structures……Page 256
12 Applications of Asymptotic Hodge theory……Page 291
13 Perverse Sheaves and D-Modules……Page 302
14 Mixed Hodge Modules……Page 338
Part V Appendices……Page 373
A Homological Algebra……Page 374
B Algebraic and Di erential Topology……Page 403
C Stratified Spaces and Singularities……Page 431
References……Page 443
Index of Notations……Page 455
Index……Page 458