Contributors. Preface

Transcription

1 Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani Complex Manifolds Definition and Examples Holomorphic Vector Bundles Differential Forms on Complex Manifolds Almost Complex Manifolds Tangent and Cotangent Space De Rham and Dolbeault Cohomologies Symplectic, Hermitian, and Kähler Structures Kähler Manifolds The Chern Class of a Holomorphic Line Bundle Harmonic Forms Hodge Theorem Compact Real Manifolds The -Laplacian Cohomology of Compact Kähler Manifolds The Kähler Identities The Hodge Decomposition Theorem Lefschetz Theorems and Hodge Riemann Bilinear Relations. 39 A Linear Algebra 45 A.1 Real and Complex Vector Spaces A.2 The Weight Filtration of a Nilpotent Transformation A.3 Representations of sl(2, C) and Lefschetz Theorems A.4 Hodge Structures B The Kähler Identities by P. A. Griffiths 58 B.1 Symplectic Linear Algebra B.2 Compatible Inner Products B.3 Symplectic Manifolds B.4 The Kähler Identities Bibliography 67

2 viii CONTENTS 2 The Algebraic de Rham Theorem by F. El Zein and L. Tu 70 Introduction Part I. Sheaf Cohomology, Hypercohomology, and the Projective Case Sheaves The Étalé Space of a Presheaf Exact Sequences of Sheaves Resolutions Sheaf Cohomology Godement s Canonical Resolution Cohomology with Coefficients in a Sheaf Flasque Sheaves Cohomology Sheaves and Exact Functors Fine Sheaves Cohomology with Coefficients in a Fine Sheaf Coherent Sheaves and Serre s GAGA Principle The Hypercohomology of a Complex of Sheaves The Spectral Sequences of Hypercohomology Acyclic Resolutions The Analytic de Rham Theorem The Holomorphic Poincaré Lemma The Analytic de Rham Theorem The Algebraic de Rham Theorem for a Projective Variety Part II. Čech Cohomology and the Algebraic de Rham Theorem in General Čech Cohomology of a Sheaf Čech Cohomology of an Open Cover Relation Between Čech Cohomology and Sheaf Cohomology Čech Cohomology of a Complex of Sheaves The Relation Between Čech Cohomology and Hypercohomology Reduction to the Affine Case Proof that the General Case Implies the Affine Case Proof that the Affine Case Implies the General Case The Algebraic de Rham Theorem for an Affine Variety The Hypercohomology of the Direct Image of a Sheaf of Smooth Forms The Hypercohomology of Rational and Meromorphic Forms Comparison of Meromorphic and Smooth Forms Bibliography Mixed Hodge Structures by F. El Zein and Lê D. T Hodge Structure on a Smooth Compact Complex Variety Hodge Structure (HS) Spectral Sequence of a Filtered Complex

3 CONTENTS ix Hodge Structure on the Cohomology of Nonsingular Compact Complex Algebraic Varieties Lefschetz Decomposition and Polarized Hodge Structure Examples Cohomology Class of a Subvariety and Hodge Conjecture Mixed Hodge Structures (MHS) Filtrations Mixed Hodge Structures (MHS) Induced Filtrations on Spectral Sequences MHS of a Normal Crossing Divisor (NCD) Mixed Hodge Complex Derived Category Derived Functor on a Filtered Complex Mixed Hodge Complex (MHC) Relative Cohomology and the Mixed Cone MHS on the Cohomology of a Complex Algebraic Variety MHS on the Cohomology of Smooth Algebraic Varieties MHS on Cohomology of Simplicial Varieties MHS on the Cohomology of a Complete Embedded Algebraic Variety Bibliography Period Domains by J. Carlson Period Domains and Monodromy Elliptic Curves Period Mappings: An Example Hodge Structures of Weight Hodge Structures of Weight Poincaré Residues Properties of the Period Mapping The Jacobian Ideal and the Local Torelli Theorem The Horizontal Distribution Distance-Decreasing Properties The Horizontal Distribution Integral Manifolds Bibliography Hodge Theory of Maps, Part I by L. Migliorini Lecture 1: The Smooth Case: E 2 -Degeneration Lecture 2: Mixed Hodge Structures Mixed Hodge Structures on the Cohomology of Algebraic Varieties The Global Invariant Cycle Theorem Semisimplicity of Monodromy

4 x CONTENTS 5.3 Lecture 3: Two Classical Theorems on Surfaces and the Local Invariant Cycle Theorem Homological Interpretation of the Contraction Criterion and Zariski s Lemma The Local Invariant Cycle Theorem, the Limit Mixed Hodge Structure, and the Clemens Schmid Exact Sequence Bibliography Hodge Theory of Maps, Part II by M. A. de Cataldo Lecture Sheaf Cohomology and All That (A Minimalist Approach) The Intersection Cohomology Complex Verdier Duality Lecture The Decomposition Theorem (DT) The Relative Hard Lefschetz and the Hard Lefschetz for Intersection Cohomology Groups Bibliography Variations of Hodge Structure by E. Cattani Local Systems and Flat Connections Local Systems Flat Bundles Analytic Families The Kodaira Spencer Map Variations of Hodge Structure Geometric Variations of Hodge Structure Abstract Variations of Hodge Structure Classifying Spaces Mixed Hodge Structures and the Orbit Theorems Nilpotent Orbits Mixed Hodge Structures SL 2 -Orbits Asymptotic Behavior of a Period Mapping Bibliography Variations of Mixed Hodge Structure by P. Brosnan and F. El Zein Variation of Mixed Hodge Structures Local Systems and Representations of the Fundamental Group Connections and Local Systems Variation of Mixed Hodge Structure of Geometric Origin Singularities of Local Systems

5 CONTENTS xi 8.2 Degeneration of Variations of Mixed Hodge Structures Diagonal Degeneration of Geometric VMHS Filtered Mixed Hodge Complex (FMHC) Diagonal Direct Image of a Simplicial Cohomological FMHC Construction of a Limit MHS on the Unipotent Nearby Cycles Case of a Smooth Morphism Polarized Hodge Lefschetz Structure Quasi-projective Case Alternative Construction, Existence and Uniqueness Admissible Variation of Mixed Hodge Structure Definition and Results Local Study of Infinitesimal Mixed Hodge Structures After Kashiwara Deligne Hodge Theory on the Cohomology of a Smooth Variety Admissible Normal Functions Reducing Theorem to a Special Case Examples Classifying Spaces Pure Classifying Spaces Mixed Classifying Spaces Local Normal Form Splittings A Formula for the Zero Locus of a Normal Function Proof of Theorem for Curves An Example Bibliography Algebraic Cycles and Chow Groups by J. Murre Lecture I: Algebraic Cycles. Chow Groups Assumptions and Conventions Algebraic Cycles Adequate Equivalence Relations Rational Equivalence. Chow Groups Lecture II: Equivalence Relations. Short Survey on the Results for Divisors Algebraic Equivalence (Weil, 1952) Smash-Nilpotent Equivalence Homological Equivalence Numerical Equivalence Final Remarks and Résumé of Relations and Notation Cartier Divisors and the Picard Group Résumé of the Main Facts for Divisors References for Lectures I and II Lecture III: Cycle Map. Intermediate Jacobian. Deligne Cohomology 425

6 xii CONTENTS The Cycle Map Hodge Classes. Hodge Conjecture Intermediate Jacobian and Abel Jacobi Map Deligne Cohomology. Deligne Cycle Map References for Lecture III Lecture IV: Algebraic Versus Homological Equivalence. Griffiths Group Lefschetz Theory Return to the Griffiths Theorem References for Lecture IV Lecture V: The Albanese Kernel. Results of Mumford, Bloch, and Bloch Srinivas The Result of Mumford Reformulation and Generalization by Bloch A Result on the Diagonal References for Lecture V Bibliography Spreads and Algebraic Cycles by M. L. Green Introduction to Spreads Cycle Class and Spreads The Conjectural Filtration on Chow Groups from a Spread Perspective The Case ofx Defined over Q The Tangent Space to Algebraic Cycles Bibliography Absolute Hodge Classes by F. Charles and C. Schnell Algebraic de Rham Cohomology Algebraic de Rham Cohomology Cycle Classes Absolute Hodge Classes Algebraic Cycles and the Hodge Conjecture Galois Action, Algebraic de Rham Cohomology, and Absolute Hodge Classes Variations on the Definition and Some Functoriality Properties Classes Coming from the Standard Conjectures and Polarizations Absolute Hodge Classes and the Hodge Conjecture Absolute Hodge Classes in Families The Variational Hodge Conjecture and the Global Invariant Cycle Theorem Deligne s Principle B The Locus of Hodge Classes Galois Action on Relative de Rham Cohomology

7 CONTENTS xiii The Field of Definition of the Locus of Hodge Classes The Kuga Satake Construction Recollection on Spin Groups Spin Representations Hodge Structures and the Deligne Torus From Weight 2 to Weight The Kuga Satake Correspondence Is Absolute Deligne s Theorem on Hodge Classes on Abelian Varieties Overview Hodge Structures of CM-Type Reduction to Abelian Varieties of CM-Type Background on Hermitian Forms Construction of Split Weil Classes André s Theorem and Reduction to Split Weil Classes Split Weil Classes are Absolute Bibliography Shimura Varieties by M. Kerr 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 538 D. Hodge-Theoretic Interpretation Locally Symmetric Varieties Complex Multiplication A. CM-Abelian Varieties B. Class Field Theory C. Main Theorem of CM Shimura Varieties A. Three Key Adélic Lemmas B. Shimura Data C. The Adélic Reformulation D. Examples Fields of Definition A. Reflex Field of a Shimura Datum B. Canonical Models C. Connected Components and VHS Bibliography 574 Index 577