Mixed Hodge Structures (eBook)

Preface 5
Contents 6
Introduction 13
Part I Basic Hodge Theory 21
1 Compact K ahler Manifolds 22
1.1 Classical Hodge Theory 22
1.2 The Lefschetz Decomposition 31
1.3 Applications 39
2 Pure Hodge Structures 44
2.1 Hodge Structures 44
2.2 Mumford-Tate Groups of Hodge Structures 51
2.3 Hodge Filtration and Hodge Complexes 54
2.4 Re ned Fundamental Classes 62
2.5 Almost K ahler V -Manifolds 67
3 Abstract Aspects of Mixed Hodge Structures 72
3.1 Introduction to Mixed Hodge Structures: Formal Aspects 73
3.2 Comparison of Filtrations 77
3.3 Mixed Hodge Structures and Mixed Hodge Complexes 80
3.4 The Mixed Cone 87
3.5 Extensions of Mixed Hodge Structures 90
Part II Mixed Hodge structures on Cohomology Groups 97
4 Smooth Varieties 98
4.1 Main Result 98
4.2 Residue Maps 101
4.3 Associated Mixed Hodge Complexes of Sheaves 105
4.4 Logarithmic Structures 108
4.5 Independence of the Compacti cation and Further Complements 110
5 Singular Varieties 118
5.1 Simplicial and Cubical Sets 118
5.2 Construction of Cubical Hyperresolutions 128
5.3 Mixed Hodge Theory for Singular Varieties 133
5.4 Cup Product and the K unneth Formula. 142
5.5 Relative Cohomology 144
6 Singular Varieties: Complementary Results 149
6.1 The Leray Filtration 149
6.2 Deleted Neighbourhoods of Algebraic Sets 152
6.3 Cup and Cap Products, and Duality 160
7 Applications to Algebraic Cycles and to Singularities 168
7.1 The Hodge Conjectures 168
7.2 Deligne Cohomology 175
7.3 The Filtered De Rham Complex And Applications 180
Part III Mixed Hodge Structures on Homotopy Groups 195
8 Hodge Theory and Iterated Integrals 196
8.1 Some Basic Results from Homotopy Theory 197
8.2 Formulation of the Main Results 201
8.3 Loop Space Cohomology and the Homotopy De Rham Theorem 204
8.4 The Homotopy De Rham Theorem for the Fundamental Group 210
8.5 Mixed Hodge Structure on the Fundamental Group 213
8.6 The Sullivan Construction 216
8.7 Mixed Hodge Structures on the Higher Homotopy Groups 218
9 Hodge Theory and Minimal Models 223
9.1 Minimal Models of Di erential Graded Algebras 224
9.2 Postnikov Towers and Minimal Models the Simply Connected Case
9.3 Mixed Hodge Structures on the Minimal Model 228
9.4 Formality of Compact K ahler Manifolds 234
Part IV Hodge Structures and Local Systems 241
10 Variations of Hodge Structure 242
10.1 Preliminaries: Local Systems over Complex Manifolds 242
10.2 Abstract Variations of Hodge Structure 244
10.3 Big Monodromy Groups, an Application 248
10.4 Variations of Hodge Structures Coming From Smooth Families 251
11 Degenerations of Hodge Structures 256
11.1 Local Systems Acquiring Singularities 256
11.2 The Limit Mixed Hodge Structure on Nearby Cycle Spaces 262
11.3 Geometric Consequences for Degenerations 277
11.4 Examples 288
12 Applications of Asymptotic Hodge theory 291
12.1 Applications to Singularities 291
12.2 An Application to Cycles: Grothendieck's Induction Principle 297
13 Perverse Sheaves and D-Modules 302
13.1 Verdier Duality 302
13.2 Perverse Complexes 307
13.3 Introduction to D-Modules 314
13.4 Coherent D-Modules 321
13.5 Filtered D-modules 328
13.6 Holonomic D-Modules 330
14 Mixed Hodge Modules 338
14.1 An Axiomatic Introduction 339
14.2 The Kashiwara-Malgrange Filtration 348
14.3 Polarizable Hodge Modules 354
14.4 Mixed Hodge Modules 363
Part V Appendices 373
A Homological Algebra 374
A.1 Additive and Abelian Categories 374
A.2 Derived Categories 379
A.3 Spectral Sequences and Filtrations 393
B Algebraic and Di erential Topology 403
B.1 Singular (Co)homology and Borel-Moore Homology 403
B.2 Sheaf Cohomology 408
B.3 Local Systems and Their Cohomology 425
C Strati ed Spaces and Singularities 431
C.1 Strati ed Spaces 431
C.2 Fibrations, and the Topology of Singularities 435
References 443
Index of Notations 455
Index 458