Hodge Theory and Complex Algebraic Geometry I 1st Edition by Claire Voisin, Leila Schneps ISBN 0521718015 9780521718011

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ISBN 10: 0521718015 
ISBN 13: 9780521718011
Author: Claire Voisin, Leila Schneps

Hodge Theory and Complex Algebraic Geometry I 1st Table of contents:

Part I: Preliminaries

1. Holomorphic Functions of Many Variables
1.1 Holomorphic functions of one variable
 1.1.1 Definition and basic properties
 1.1.2 Background on Stokes’ formula
 1.1.3 Cauchy’s formula
1.2 Holomorphic functions of several variables
 1.2.1 Cauchy’s formula and analyticity
 1.2.2 Applications of Cauchy’s formula
1.3 The equation…
Exercises

2. Complex Manifolds
2.1 Manifolds and vector bundles
 2.1.1 Definitions
 2.1.2 The tangent bundle
 2.1.3 Complex manifolds
2.2 Integrability of almost complex structures
 2.2.1 Tangent bundle of a complex manifold
 2.2.2 The Frobenius theorem
 2.2.3 The Newlander–Nirenberg theorem
2.3 The operators…
 2.3.1 Definition
 2.3.2 Local exactness
 2.3.3 Dolbeault complex of a holomorphic bundle
2.4 Examples of complex manifolds
 - Riemann surfaces
 - Complex projective space
 - Complex tori
Exercises

3. Kähler Metrics
3.1 Definition and basic properties
 3.1.1 Hermitian geometry
 3.1.2 Hermitian and Kähler metrics
 3.1.3 Basic properties
  - Volume form
  - Submanifolds
3.2 Characterisations of Kähler metrics
 3.2.1 Background on connections
 3.2.2 Kähler metrics and connections
3.3 Examples of Kähler manifolds
 3.3.1 Chern form of line bundles
 3.3.2 Fubini–Study metric
 3.3.3 Blowups
Exercises

4. Sheaves and Cohomology
4.1 Sheaves
 4.1.1 Definitions, examples
 4.1.2 Stalks, kernels, images
 4.1.3 Resolutions
  - The Cech resolution
  - The de Rham resolution
  - The Dolbeault resolution
4.2 Functors and derived functors
 4.2.1 Abelian categories
 4.2.2 Injective resolutions
 4.2.3 Derived functors
4.3 Sheaf cohomology
 4.3.1 Acyclic resolutions
 4.3.2 The de Rham theorems
 4.3.3 Interpretations of the group H
Exercises

Part II: The Hodge Decomposition

5. Harmonic Forms and Cohomology
5.1 Laplacians
 5.1.1 The L metric
 5.1.2 Formal adjoint operators
 5.1.3 Adjoints of the operators…
 5.1.4 Laplacians
5.2 Elliptic differential operators
 5.2.1 Symbols of differential operators
 5.2.2 Symbol of the Laplacian
 5.2.3 The fundamental theorem
5.3 Applications
 5.3.1 Cohomology and harmonic forms
 5.3.2 Duality theorems
Exercises

6. The Case of Kähler Manifolds
6.1 The Hodge decomposition
 6.1.1 Kähler identities
 6.1.2 Comparison of the Laplacians
 6.1.3 Other applications
6.2 Lefschetz decomposition
 6.2.1 Commutators
 6.2.2 Lefschetz decomposition on forms
 6.2.3 Lefschetz decomposition on the cohomology
6.3 The Hodge index theorem
 6.3.1 Other Hermitian identities
 6.3.2 The Hodge index theorem
Exercises

7. Hodge Structures and Polarisations
7.1 Definitions, basic properties
 7.1.1 Hodge structure
 7.1.2 Polarisation
 7.1.3 Polarised varieties
7.2 Examples
 7.2.1 Projective space
 7.2.2 Hodge structures of weight 1 and abelian varieties
 7.2.3 Hodge structures of weight 2
7.3 Functoriality
 7.3.1 Morphisms of Hodge structures
 7.3.2 The pullback and the Gysin morphism
 7.3.3 Hodge structure of a blowup
Exercises

8. Holomorphic de Rham Complexes and Spectral Sequences
8.1 Hypercohomology
 8.1.1 Resolutions of complexes
 8.1.2 Derived functors
 8.1.3 Composed functors
  - Application: Proof of the Leray–Hirsch theorem 7.33
8.2 Holomorphic de Rham complexes
 8.2.1 Holomorphic de Rham resolutions
 8.2.2 The logarithmic case
 8.2.3 Cohomology of the logarithmic complex
8.3 Filtrations and spectral sequences
 8.3.1 Filtered complexes
 8.3.2 Spectral sequences
 8.3.3 The Frölicher spectral sequence
8.4 Hodge theory of open manifolds
 8.4.1 Filtrations on the logarithmic complex
 8.4.2 First terms of the spectral sequence
 8.4.3 Deligne’s theorem
Exercises

Part III: Variations of Hodge Structure

9. Families and Deformations
9.1 Families of manifolds
 9.1.1 Trivialisations
 9.1.2 The Kodaira–Spencer map
9.2 The Gauss–Manin connection
 9.2.1 Local systems and flat connections
 9.2.2 The Cartan–Lie formula
9.3 The Kähler case
 9.3.1 Semicontinuity theorems
 9.3.2 The Hodge numbers are constant
 9.3.3 Stability of Kähler manifolds

10. Variations of Hodge Structure
10.1 Period domain and period map
 10.1.1 Grassmannians
 10.1.2 The period map
 10.1.3 The period domain
10.2 Variations of Hodge structure
 10.2.1 Hodge bundles
 10.2.2 Transversality
 10.2.3 Computation of the differential
10.3 Applications
 10.3.1 Curves
 10.3.2 Calabi–Yau manifolds
Exercises

Part IV: Cycles and Cycle Classes

11. Hodge Classes
11.1 Cycle class
 11.1.1 Analytic subsets
 11.1.2 Cohomology class
 11.1.3 The Kähler case
 11.1.4 Other approaches
11.2 Chern classes
 11.2.1 Construction
 11.2.2 The Kähler case
11.3 Hodge classes
 11.3.1 Definitions and examples
 11.3.2 The Hodge conjecture
 11.3.3 Correspondences
Exercises

12. Deligne–Beilinson Cohomology and the Abel–Jacobi Map
12.1 The Abel–Jacobi map
 12.1.1 Intermediate Jacobians
 12.1.2 The Abel–Jacobi map
 12.1.3 Picard and Albanese varieties
12.2 Properties
 12.2.1 Correspondences
 12.2.2 Some results
12.3 Deligne cohomology
 12.3.1 The Deligne complex
 12.3.2 Differential characters
 12.3.3 Cycle class

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Tags: Claire Voisin, Leila Schneps, Hodge Theory, Algebraic Geometry