Mixed hodge structures /

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Bibliographic Details
Author / Creator:	Peters, C. (Chris)
Imprint:	Berlin : Springer, c2008.
Description:	xiii, 470 p. : ill. ; 24 cm.
Language:	English
Series:	Ergebnisse der Mathematik und ihrer Grenzgebiete = Series of modern surveys in mathematics ; 3. Folge, Bd. 52 Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 52.
Subject:	Hodge theory. Hodge theory.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/6825821

Hidden Bibliographic Details
Other authors / contributors:	Steenbrink, J. H. M.
ISBN:	3540770151 9783540770152
Notes:	Includes bibliographical references (p. [445]-456) and indexes.

Table of Contents:

Introduction
Part I. Basic Hodge Theory
1. Compact Kahler Manifolds
1.1. Classical Hodge Theory
1.1.1. Harmonic Theory
1.1.2. The Hodge Decomposition
1.1.3. Hodge Structures in Cohomology and Homology
1.2. The Lefschetz Decomposition
1.2.1. Representation Theory of SL(2, R)
1.2.2. Primitive Cohomology
1.3. Applications
2. Pure Hodge Structures
2.1. Hodge Structures
2.1.1. Basic Definitions
2.1.2. Polarized Hodge Structures
2.2. Mumford-Tate Groups of Hodge Structures
2.3. Hodge Filtration and Hodge Complexes
2.3.1. Hodge to De Rham Spectral Sequence
2.3.2. Strong Hodge Decompositions
2.3.3. Hodge Complexes and Hodge Complexes of Sheaves
2.4. Refined Fundamental Classes
2.5. Almost Kahler V-Manifolds
3. Abstract Aspects of Mixed Hodge Structures
3.1. Introduction to Mixed Hodge Structures: Formal Aspects
3.2. Comparison of Filtrations
3.3. Mixed Hodge Structures and Mixed Hodge Complexes
3.4. The Mixed Cone
3.5. Extensions of Mixed Hodge Structures
3.5.1. Mixed Hodge Extensions
3.5.2. Iterated Extensions and Absolute Hodge Cohomology
Part II. Mixed Hodge structures on Cohomology Groups
4. Smooth Varieties
4.1. Main Result
4.2. Residue Maps
4.3. Associated Mixed Hodge Complexes of Sheaves
4.4. Logarithmic Structures
4.5. Independence of the Compactification and Further Complements
4.5.1. Invariance
4.5.2. Restrictions for the Hodge Numbers
4.5.3. Theorem of the Fixed Part and Applications
4.5.4. Application to Lefschetz Pencils
5. Singular Varieties
5.1. Simplicial and Cubical Sets
5.1.1. Basic Definitions
5.1.2. Sheaves on Semi-simplicial Spaces and Their Cohomology
5.1.3. Cohomological Descent and Resolutions
5.2. Construction of Cubical Hyperresolutions
5.3. Mixed Hodge Theory for Singular Varieties
5.3.1. The Basic Construction
5.3.2. Mixed Hodge Theory of Proper Modifications
5.3.3. Restriction on the Hodge Numbers
5.4. Cup Product and the Kunneth Formula
5.5. Relative Cohomology
5.5.1. Construction of the Mixed Hodge Structure
5.5.2. Cohomology with Compact Support
6. Singular Varieties: Complementary Results
6.1. The Leray Filtration
6.2. Deleted Neighbourhoods of Algebraic Sets
6.2.1. Mixed Hodge Complexes
6.2.2. Products and Deleted Neighbourhoods
6.2.3. Semi-purity of the Link
6.3. Cup and Cap Products, and Duality
6.3.1. Duality for Cohomology with Compact Supports
6.3.2. The Extra-Ordinary Cup Product
7. Applications to Algebraic Cycles and to Singularities
7.1. The Hodge Conjectures
7.1.1. Versions for Smooth Projective Varieties
7.1.2. The Hodge Conjecture and the Intermediate Jacobian
7.1.3. A Version for Singular Varieties
7.2. Deligne Cohomology
7.2.1. Basic Properties
7.2.2. Cycle Classes for Deligne Cohomology
7.3. The Filtered De Rham Complex And Applications
7.3.1. The Filtered De Rham Complex
7.3.2. Application to Vanishing Theorems
7.3.3. Applications to Du Bois Singularities
Part III. Mixed Hodge Structures on Homotopy Groups
8. Hodge Theory and Iterated Integrals
8.1. Some Basic Results from Homotopy Theory
8.2. Formulation of the Main Results
8.3. Loop Space Cohomology and the Homotopy De Rham Theorem
8.3.1. Iterated Integrals
8.3.2. Chen's Version of the De Rham Theorem
8.3.3. The Bar Construction
8.3.4. Iterated Integrals of 1-Forms
8.4. The Homotopy De Rham Theorem for the Fundamental Group
8.5. Mixed Hodge Structure on the Fundamental Group
8.6. The Sullivan Construction
8.7. Mixed Hodge Structures on the Higher Homotopy Groups
9. Hodge Theory and Minimal Models
9.1. Minimal Models of Differential Graded Algebras
9.2. Postnikov Towers and Minimal Models; the Simply Connected Case
9.3. Mixed Hodge Structures on the Minimal Model
9.4. Formality of Compact Kahler Manifolds
9.4.1. The 1-Minimal Model
9.4.2. The De Rham Fundamental Group
9.4.3. Formality
Part IV. Hodge Structures and Local Systems
10. Variations of Hodge Structure
10.1. Preliminaries: Local Systems over Complex Manifolds
10.2. Abstract Variations of Hodge Structure
10.3. Big Monodromy Groups, an Application
10.4. Variations of Hodge Structures Coming From Smooth Families
11. Degenerations of Hodge Structures
11.1. Local Systems Acquiring Singularities
11.1.1. Connections with Logarithmic Poles
11.1.2. The Riemann-Hilbert Correspondence (I)
11.2. The Limit Mixed Hodge Structure on Nearby Cycle Spaces
11.2.1. Asymptotics for Variations of Hodge Structure over a Punctured Disk
11.2.2. Geometric Set-Up and Preliminary Reductions
11.2.3. The Nearby and Vanishing Cycle Functor
11.2.4. The Relative Logarithmic de Rham Complex and Quasi-unipotency of the Monodromy
11.2.5. The Complex Monodromy Weight Filtration and the Hodge Filtration
11.2.6. The Rational Structure
11.2.7. The Mixed Hodge Structure on the Limit
11.3. Geometric Consequences for Degenerations
11.3.1. Monodromy, Specialization and Wang Sequence
11.3.2. The Monodromy and Local Invariant Cycle Theorems
11.4. Examples
12. Applications of Asymptotic Hodge theory
12.1. Applications to Singularities
12.1.1. Localizing Nearby Cycles
12.1.2. A Mixed Hodge Structure on the Cohomology of Milnor Fibres
12.1.3. The Spectrum of Singularities
12.2. An Application to Cycles: Grothendieck's Induction Principle
13. Perverse Sheaves and D-Modules
13.1. Verdier Duality
13.1.1. Dimension
13.1.2. The Dualizing Complex
13.1.3. Statement of Verdier Duality
13.1.4. Extraordinary Pull Back
13.2. Perverse Complexes
13.2.1. Intersection Homology and Cohomology
13.2.2. Constructible and Perverse Complexes
13.2.3. An Example: Nearby and Vanishing Cycles
13.3. Introduction to D-Modules
13.3.1. Integrable Connections and D-Modules
13.3.2. From Left to Right and Vice Versa
13.3.3. Derived Categories of D-modules
13.3.4. Inverse and Direct Images
13.3.5. An Example: the Gauss-Manin System
13.4. Coherent D-Modules
13.4.1. Basic Definitions
13.4.2. Good Filtrations and Characteristic Varieties
13.4.3. Behaviour under Direct and Inverse Images
13.5. Filtered D-modules
13.5.1. Derived Categories
13.5.2. Duality
13.5.3. Functoriality
13.6. Holonomic D-Modules
13.6.1. Symplectic Geometry
13.6.2. Basics on Holonomic D-Modules
13.6.3. The Riemann-Hilbert Correspondence (II)
14. Mixed Hodge Modules
14.1. An Axiomatic Introduction
14.1.1. The Axioms
14.1.2. First Consequences of the Axioms
14.1.3. Spectral Sequences
14.1.4. Intersection Cohomology
14.1.5. Refined Fundamental Classes
14.2. The Kashiwara-Malgrange Filtration
14.2.1. Motivation
14.2.2. The Rational V-Filtration
14.3. Polarizable Hodge Modules
14.3.1. Hodge Modules
14.3.2. Polarizations
14.3.3. Lefschetz Operators and the Decomposition Theorem
14.4. Mixed Hodge Modules
14.4.1. Variations of Mixed Hodge Structure
14.4.2. Defining Mixed Hodge Modules
14.4.3. About the Axioms
14.4.4. Application: Vanishing Theorems
14.4.5. The Motivic Hodge Character and Motivic Chern Classes
Part V. Appendices
A. Homological Algebra
A.1. Additive and Abelian Categories
A.1.1. Pre-Abelian Categories
A.1.2. Additive Categories
A.2. Derived Categories
A.2.1. The Homotopy Category
A.2.2. The Derived Category
A.2.3. Injective and Projective Resolutions
A.2.4. Derived Functors
A.2.5. Properties of the Ext-functor
A.2.6. Yoneda Extensions
A.3. Spectral Sequences and Filtrations
A.3.1. Filtrations
A.3.2. Spectral Sequences and Exact Couples
A.3.3. Filtrations Induce Spectral Sequences
A.3.4. Derived Functors and Spectral Sequences
B. Algebraic and Differential Topology
B.1. Singular (Co)homology and Borel-Moore Homology
B.1.1. Basic Definitions and Tools
B.1.2. Pairings and Products
B.2. Sheaf Cohomology
B.2.1. The Godement Resolution and Cohomology
B.2.2. Cohomology and Supports
B.2.3. Cech Cohomology
B.2.4. De Rham Theorems
B.2.5. Direct and Inverse Images
B.2.6. Sheaf Cohomology and Closed Subspaces
B.2.7. Mapping Cones and Cylinders
B.2.8. Duality Theorems on Manifolds
B.2.9. Orientations and Fundamental Classes
B.3. Local Systems and Their Cohomology
B.3.1. Local Systems and Locally Constant Sheaves
B.3.2. Homology and Cohomology
B.3.3. Local Systems and Flat Connections
C. Stratified Spaces and Singularities
C.1. Stratified Spaces
C.1.1. Pseudomanifolds
C.1.2. Whitney Stratifications
C.2. Fibrations, and the Topology of Singularities
C.2.1. The Milnor Fibration
C.2.2. Topology of One-parameter Degenerations
C.2.3. An Example: Lefschetz Pencils
References
Index of Notations
Index

Mixed hodge structures /

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