Mixed Hodge Structures

by Peters, Chris A. M.; Steenbrink, Joseph H. M.

• ISBN13: 9783540770152
• ISBN10: 3540770151
• Format: Hardcover
• Copyright: 2008-04-01
• Publisher: Springer Verlag

Summary

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

Table of Contents

Introduction	p. 1
Basic Hodge Theory
Compact Kahler Manifolds	p. 11
Classical Hodge Theory	p. 11
Harmonic Theory	p. 11
The Hodge Decomposition	p. 15
Hodge Structures in Cohomology and Homology	p. 17
The Lefschetz Decomposition	p. 20
Representation Theory of SL(2, R)	p. 20
Primitive Cohomology	p. 24
Applications	p. 28
Pure Hodge Structures	p. 33
Hodge Structures	p. 33
Basic Definitions	p. 33
Polarized Hodge Structures	p. 38
Mumford-Tate Groups of Hodge Structures	p. 40
Hodge Filtration and Hodge Complexes	p. 43
Hodge to De Rham Spectral Sequence	p. 43
Strong Hodge Decompositions	p. 45
Hodge Complexes and Hodge Complexes of Sheaves	p. 49
Refined Fundamental Classes	p. 51
Almost Kahler V-Manifolds	p. 56
Abstract Aspects of Mixed Hodge Structures	p. 61
Introduction to Mixed Hodge Structures: Formal Aspects	p. 62
Comparison of Filtrations	p. 66
Mixed Hodge Structures and Mixed Hodge Complexes	p. 69
The Mixed Cone	p. 76
Extensions of Mixed Hodge Structures	p. 79
Mixed Hodge Extensions	p. 79
Iterated Extensions and Absolute Hodge Cohomology	p. 83
Mixed Hodge structures on Cohomology Groups
Smooth Varieties	p. 89
Main Result	p. 89
Residue Maps	p. 92
Associated Mixed Hodge Complexes of Sheaves	p. 96
Logarithmic Structures	p. 99
Independence of the Compactification and Further Complements	p. 101
Invariance	p. 101
Restrictions for the Hodge Numbers	p. 102
Theorem of the Fixed Part and Applications	p. 103
Application to Lefschetz Pencils	p. 105
Singular Varieties	p. 109
Simplicial and Cubical Sets	p. 109
Basic Definitions	p. 109
Sheaves on Semi-simplicial Spaces and Their Cohomology	p. 114
Cohomological Descent and Resolutions	p. 117
Construction of Cubical Hyperresolutions	p. 119
Mixed Hodge Theory for Singular Varieties	p. 124
The Basic Construction	p. 124
Mixed Hodge Theory of Proper Modifications	p. 128
Restriction on the Hodge Numbers	p. 130
Cup Product and the Kunneth Formula	p. 133
Relative Cohomology	p. 135
Construction of the Mixed Hodge Structure	p. 135
Cohomology with Compact Support	p. 137
Singular Varieties: Complementary Results	p. 141
The Leray Filtration	p. 141
Deleted Neighbourhoods of Algebraic Sets	p. 144
Mixed Hodge Complexes	p. 144
Products and Deleted Neighbourhoods	p. 146
Semi-purity of the Link	p. 150
Cup and Cap Products, and Duality	p. 152
Duality for Cohomology with Compact Supports	p. 152
The Extra-Ordinary Cup Product	p. 156
Applications to Algebraic Cycles and to Singularities	p. 161
The Hodge Conjectures	p. 161
Versions for Smooth Projective Varieties	p. 161
The Hodge Conjecture and the Intermediate Jacobian	p. 164
A Version for Singular Varieties	p. 166
Deligne Cohomology	p. 168
Basic Properties	p. 168
Cycle Classes for Deligne Cohomology	p. 172
The Filtered De Rham Complex And Applications	p. 173
The Filtered De Rham Complex	p. 173
Application to Vanishing Theorems	p. 178
Applications to Du Bois Singularities	p. 183
Mixed Hodge Structures on Homotopy Groups
Hodge Theory and Iterated Integrals	p. 191
Some Basic Results from Homotopy Theory	p. 192
Formulation of the Main Results	p. 196
Loop Space Cohomology and the Homotopy De Rham Theorem	p. 199
Iterated Integrals	p. 199
Chen's Version of the De Rham Theorem	p. 201
The Bar Construction	p. 202
Iterated Integrals of 1-Forms	p. 204
The Homotopy De Rham Theorem for the Fundamental Group	p. 205
Mixed Hodge Structure on the Fundamental Group	p. 208
The Sullivan Construction	p. 211
Mixed Hodge Structures on the Higher Homotopy Groups	p. 213
Hodge Theory and Minimal Models	p. 219
Minimal Models of Differential Graded Algebras	p. 220
Postnikov Towers and Minimal Models; the Simply Connected Case	p. 222
Mixed Hodge Structures on the Minimal Model	p. 224
Formality of Compact Kahler Manifolds	p. 230
The 1-Minimal Model	p. 230
The De Rham Fundamental Group	p. 232
Formality	p. 234
Hodge Structures and Local Systems
Variations of Hodge Structure	p. 239
Preliminaries: Local Systems over Complex Manifolds	p. 239
Abstract Variations of Hodge Structure	p. 241
Big Monodromy Groups, an Application	p. 245
Variations of Hodge Structures Coming From Smooth Families	p. 248
Degenerations of Hodge Structures	p. 253
Local Systems Acquiring Singularities	p. 253
Connections with Logarithmic Poles	p. 253
The Riemann-Hilbert Correspondence (I)	p. 256
The Limit Mixed Hodge Structure on Nearby Cycle Spaces	p. 259
Asymptotics for Variations of Hodge Structure over a Punctured Disk	p. 259
Geometric Set-Up and Preliminary Reductions	p. 260
The Nearby and Vanishing Cycle Functor	p. 262
The Relative Logarithmic de Rham Complex and Quasi-unipotency of the Monodromy	p. 263
The Complex Monodromy Weight Filtration and the Hodge Filtration	p. 268
The Rational Structure	p. 271
The Mixed Hodge Structure on the Limit	p. 272
Geometric Consequences for Degenerations	p. 274
Monodromy, Specialization and Wang Sequence	p. 274
The Monodromy and Local Invariant Cycle Theorems	p. 279
Examples	p. 285
Applications of Asymptotic Hodge theory	p. 289
Applications to Singularities	p. 289
Localizing Nearby Cycles	p. 289
A Mixed Hodge Structure on the Cohomology of Milnor Fibres	p. 291
The Spectrum of Singularities	p. 293
An Application to Cycles: Grothendieck's Induction Principle	p. 295
Perverse Sheaves and D-Modules	p. 301
Verdier Duality	p. 301
Dimension	p. 301
The Dualizing Complex	p. 302
Statement of Verdier Duality	p. 304
Extraordinary Pull Back	p. 305
Perverse Complexes	p. 306
Intersection Homology and Cohomology	p. 306
Constructible and Perverse Complexes	p. 308
An Example: Nearby and Vanishing Cycles	p. 312
Introduction to D-Modules	p. 313
Integrable Connections and D-Modules	p. 313
From Left to Right and Vice Versa	p. 315
Derived Categories of D-modules	p. 316
Inverse and Direct Images	p. 317
An Example: the Gauss-Manin System	p. 320
Coherent D-Modules	p. 320
Basic Definitions	p. 321
Good Filtrations and Characteristic Varieties	p. 323
Behaviour under Direct and Inverse Images	p. 325
Filtered D-modules	p. 327
Derived Categories	p. 327
Duality	p. 328
Functoriality	p. 328
Holonomic D-Modules	p. 329
Symplectic Geometry	p. 329
Basics on Holonomic D-Modules	p. 331
The Riemann-Hilbert Correspondence (II)	p. 332
Mixed Hodge Modules	p. 337
An Axiomatic Introduction	p. 338
The Axioms	p. 338
First Consequences of the Axioms	p. 340
Spectral Sequences	p. 343
Intersection Cohomology	p. 345
Refined Fundamental Classes	p. 347
The Kashiwara-Malgrange Filtration	p. 347
Motivation	p. 347
The Rational V-Filtration	p. 349
Polarizable Hodge Modules	p. 353
Hodge Modules	p. 353
Polarizations	p. 357
Lefschetz Operators and the Decomposition Theorem	p. 359
Mixed Hodge Modules	p. 362
Variations of Mixed Hodge Structure	p. 362
Defining Mixed Hodge Modules	p. 365
About the Axioms	p. 366
Application: Vanishing Theorems	p. 367
The Motivic Hodge Character and Motivic Chern Classes	p. 368
Appendices
Homological Algebra	p. 375
Additive and Abelian Categories	p. 375
Pre-Abelian Categories	p. 376
Additive Categories	p. 377
Derived Categories	p. 380
The Homotopy Category	p. 380
The Derived Category	p. 382
Injective and Projective Resolutions	p. 386
Derived Functors	p. 388
Properties of the Ext-functor	p. 391
Yoneda Extensions	p. 391
Spectral Sequences and Filtrations	p. 394
Filtrations	p. 394
Spectral Sequences and Exact Couples	p. 397
Filtrations Induce Spectral Sequences	p. 398
Derived Functors and Spectral Sequences	p. 401
Algebraic and Differential Topology	p. 405
Singular (Co)homology and Borel-Moore Homology	p. 405
Basic Definitions and Tools	p. 405
Pairings and Products	p. 409
Sheaf Cohomology	p. 410
The Godement Resolution and Cohomology	p. 410
Cohomology and Supports	p. 412
Cech Cohomology	p. 414
De Rham Theorems	p. 416
Direct and Inverse Images	p. 417
Sheaf Cohomology and Closed Subspaces	p. 420
Mapping Cones and Cylinders	p. 421
Duality Theorems on Manifolds	p. 422
Orientations and Fundamental Classes	p. 424
Local Systems and Their Cohomology	p. 427
Local Systems and Locally Constant Sheaves	p. 428
Homology and Cohomology	p. 429
Local Systems and Flat Connections	p. 430
Stratified Spaces and Singularities	p. 433
Stratified Spaces	p. 433
Pseudomanifolds	p. 433
Whitney Stratifications	p. 434
Fibrations, and the Topology of Singularities	p. 437
The Milnor Fibration	p. 437
Topology of One-parameter Degenerations	p. 438
An Example: Lefschetz Pencils	p. 441
References	p. 445
Index of Notations	p. 457
Index	p. 461
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