Chapter
2.1.2 Exact Sequences of Sheaves
2.2.1 Godement’s Canonical Resolution
2.2.2 Cohomology with Coefficients in a Sheaf
2.2.4 Cohomology Sheaves and Exact Functors
2.2.6 Cohomology with Coefficients in a Fine Sheaf
2.3 Coherent Sheaves and Serre’s GAGA Principle
2.4 The Hypercohomology of a Complex of Sheaves
2.4.1 The Spectral Sequences of Hypercohomology
2.4.2 Acyclic Resolutions
2.5 The Analytic de Rham Theorem
2.5.1 The Holomorphic Poincaré Lemma
2.5.2 The Analytic de Rham Theorem
2.6 The Algebraic de Rham Theorem for a Projective Variety
Part II. Čech Cohomology and the Algebraic de Rham Theorem in General
2.7 Čech Cohomology of a Sheaf
2.7.1 Čech Cohomology of an Open Cover
2.7.2 Relation Between Čech Cohomology and Sheaf Cohomology
2.8 Čech Cohomology of a Complex of Sheaves
2.8.1 The Relation Between Čech Cohomology and Hypercohomology
2.9 Reduction to the Affine Case
2.9.1 Proof that the General Case Implies the Affine Case
2.9.2 Proof that the Affine Case Implies the General Case
2.10 The Algebraic de Rham Theorem for an Affine Variety
2.10.1 The Hypercohomology of the Direct Image of a Sheaf of Smooth Forms
2.10.2 The Hypercohomology of Rational and Meromorphic Forms
2.10.3 Comparison of Meromorphic and Smooth Forms
3.1 Hodge Structure on a Smooth Compact Complex Variety
3.1.1 Hodge Structure (HS)
3.1.2 Spectral Sequence of a Filtered Complex
3.1.3 Hodge Structure on the Cohomology of Nonsingular Compact Complex Algebraic Varieties
3.1.4 Lefschetz Decomposition and Polarized Hodge Structure
3.1.6 Cohomology Class of a Subvariety and Hodge Conjecture
3.2 Mixed Hodge Structures (MHS)
3.2.2 Mixed Hodge Structures (MHS)
3.2.3 Induced Filtrations on Spectral Sequences
3.2.4 MHS of a Normal Crossing Divisor (NCD)
3.3.2 Derived Functor on a Filtered Complex
3.3.3 Mixed Hodge Complex (MHC)
3.3.4 Relative Cohomology and the Mixed Cone
3.4 MHS on the Cohomology of a Complex Algebraic Variety
3.4.1 MHS on the Cohomology of Smooth Algebraic Varieties
3.4.2 MHS on Cohomology of Simplicial Varieties
3.4.3 MHS on the Cohomology of a Complete Embedded Algebraic Variety
4.1 Period Domains and Monodromy
4.3 Period Mappings: An Example
4.4 Hodge Structures of Weight 1
4.5 Hodge Structures of Weight 2
4.7 Properties of the Period Mapping
4.8 The Jacobian Ideal and the Local Torelli Theorem
4.9 The Horizontal Distribution—Distance-Decreasing Properties
4.10 The Horizontal Distribution—Integral Manifolds
5 Hodge Theory of Maps, Part I
5.1 Lecture 1: The Smooth Case: E2-Degeneration
5.2 Lecture 2: Mixed Hodge Structures
5.2.1 Mixed Hodge Structures on the Cohomology of Algebraic Varieties
5.2.2 The Global Invariant Cycle Theorem
5.2.3 Semisimplicity of Monodromy
5.3 Lecture 3: Two Classical Theorems on Surfaces and the Local Invariant Cycle Theorem
5.3.1 Homological Interpretation of the Contraction Criterion and Zariski’s Lemma
5.3.2 The Local Invariant Cycle Theorem, the Limit Mixed Hodge Structure, and the Clemens–Schmid Exact Sequence
6 Hodge Theory of Maps, Part II
6.1.1 Sheaf Cohomology and All That (A Minimalist Approach)
6.1.2 The Intersection Cohomology Complex
6.2.1 The Decomposition Theorem (DT)
6.2.2 The Relative Hard Lefschetz and the Hard Lefschetz for Inter-section Cohomology Groups
7 Variations of Hodge Structure
7.1 Local Systems and Flat Connections
7.2.1 The Kodaira–Spencer Map
7.3 Variations of Hodge Structure
7.3.1 Geometric Variations of Hodge Structure
7.3.2 Abstract Variations of Hodge Structure
7.5 Mixed Hodge Structures and the Orbit Theorems
7.5.2 Mixed Hodge Structures
7.6 Asymptotic Behavior of a Period Mapping
8 Variations of Mixed Hodge Structure
8.1 Variation of Mixed Hodge Structures
8.1.1 Local Systems and Representations of the Fundamental Group
8.1.2 Connections and Local Systems
8.1.3 Variation of Mixed Hodge Structure of Geometric Origin
8.1.4 Singularities of Local Systems
8.2 Degeneration of Variations of Mixed Hodge Structures
8.2.1 Diagonal Degeneration of Geometric VMHS
8.2.2 Filtered Mixed Hodge Complex (FMHC)
8.2.3 Diagonal Direct Image of a Simplicial Cohomological FMHC
8.2.4 Construction of a Limit MHS on the Unipotent Nearby Cycles
8.2.5 Case of a Smooth Morphism
8.2.6 Polarized Hodge–Lefschetz Structure
8.2.7 Quasi-projective Case
8.2.8 Alternative Construction, Existence and Uniqueness
8.3 Admissible Variation of Mixed Hodge Structure
8.3.1 Definition and Results
8.3.2 Local Study of Infinitesimal Mixed Hodge Structures After Kashiwara
8.3.3 Deligne–Hodge Theory on the Cohomology of a Smooth Variety
8.4 Admissible Normal Functions
8.4.1 Reducing Theorem 8.4.6 to a Special Case
8.4.4 Pure Classifying Spaces
8.4.5 Mixed Classifying Spaces
8.4.8 A Formula for the Zero Locus of a Normal Function
8.4.9 Proof of Theorem 8.4.6 for Curves
9 Algebraic Cycles and Chow Groups
9.1 Lecture I: Algebraic Cycles. Chow Groups
9.1.1 Assumptions and Conventions
9.1.3 Adequate Equivalence Relations
9.1.4 Rational Equivalence. Chow Groups
9.2 Lecture II: Equivalence Relations. Short Survey on the Results for Divisors
9.2.1 Algebraic Equivalence (Weil, 1952)
9.2.2 Smash-Nilpotent Equivalence
9.2.3 Homological Equivalence
9.2.4 Numerical Equivalence
9.2.5 Final Remarks and Résumé of Relations and Notation
9.2.6 Cartier Divisors and the Picard Group
9.2.7 Résumé of the Main Facts for Divisors
9.2.8 References for Lectures I and II
9.3 Lecture III: Cycle Map. Intermediate Jacobian. Deligne Cohomology
9.3.2 Hodge Classes. Hodge Conjecture
9.3.3 Intermediate Jacobian and Abel–Jacobi Map
9.3.4 Deligne Cohomology. Deligne Cycle Map
9.3.5 References for Lecture III
9.4 Lecture IV: Algebraic Versus Homological Equivalence. Griffiths Group
9.4.2 Return to the Griffiths Theorem
9.4.3 References for Lecture IV
9.5 Lecture V: The Albanese Kernel. Results of Mumford, Bloch, and Bloch–Srinivas
9.5.1 The Result of Mumford
9.5.2 Reformulation and Generalization by Bloch
9.5.3 A Result on the Diagonal
9.5.4 References for Lecture V
10 Spreads and Algebraic Cycles
10.1 Introduction to Spreads
10.2 Cycle Class and Spreads
10.3 The Conjectural Filtration on Chow Groups from a Spread Perspective
10.4 The Case of X Defined over Q
10.5 The Tangent Space to Algebraic Cycles
11 Absolute Hodge Classes
11.1 Algebraic de Rham Cohomology
11.1.1 Algebraic de Rham Cohomology
11.2 Absolute Hodge Classes
11.2.1 Algebraic Cycles and the Hodge Conjecture
11.2.2 Galois Action, Algebraic de Rham Cohomology, and Absolute Hodge Classes
11.2.3 Variations on the Definition and Some Functoriality Properties
11.2.4 Classes Coming from the Standard Conjectures and Polarizations
11.2.5 Absolute Hodge Classes and the Hodge Conjecture
11.3 Absolute Hodge Classes in Families
11.3.1 The Variational Hodge Conjecture and the Global Invariant Cycle Theorem
11.3.2 Deligne’s Principle B
11.3.3 The Locus of Hodge Classes
11.3.4 Galois Action on Relative de Rham Cohomology
11.3.5 The Field of Definition of the Locus of Hodge Classes
11.4 The Kuga–Satake Construction
11.4.1 Recollection on Spin Groups
11.4.2 Spin Representations
11.4.3 Hodge Structures and the Deligne Torus
11.4.4 From Weight 2 to Weight 1
11.4.5 The Kuga–Satake Correspondence Is Absolute
11.5 Deligne’s Theorem on Hodge Classes on Abelian Varieties
11.5.2 Hodge Structures of CM-Type
11.5.3 Reduction to Abelian Varieties of CM-Type
11.5.4 Background on Hermitian Forms
11.5.5 Construction of Split Weil Classes
11.5.6 André’s Theorem and Reduction to Split Weil Classes
11.5.7 Split Weil Classes are Absolute
12.1 Hermitian Symmetric Domains
A. Algebraic Groups and Their Properties
B. Three Characterizations of Hermitian Symmetric Domains
C. Cartan’s Classification of Irreducible Hermitian Symmetric Domains
D. Hodge-Theoretic Interpretation
12.2 Locally Symmetric Varieties
12.3 Complex Multiplication
A. Three Key Adélic Lemmas
C. The Adélic Reformulation
12.5 Fields of Definition
A. Reflex Field of a Shimura Datum
C. Connected Components and VHS
Hodge Theory (MN-49)
：Hodge Theory (MN-49)
( Mathematical Notes )
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