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Hodge Theory and Complex Algebraic Geometry I

Book Description

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. 0. Introduction
  7. I: Preliminaries
    1. 1. Holomorphic Functions of Many Variables
      1. 1.1 Holomorphic functions of one variable
        1. 1.1.1 Definition and basic properties
        2. 1.1.2 Background on Stokes’ formula
        3. 1.1.3 Cauchy’s formula
      2. 1.2 Holomorphic functions of several variables
        1. 1.2.1 Cauchy’s formula and analyticity
        2. 1.2.2 Applications of Cauchy’s formula
      3. 1.3 The equation ∂g/∂z = f
      4. Exercises
    2. 2. Complex Manifolds
      1. 2.1 Manifolds and vector bundles
        1. 2.1.1 Definitions
        2. 2.1.2 The tangent bundle
        3. 2.1.3 Complex manifolds
      2. 2.2 Integrability of almost complex structures
        1. 2.2.1 Tangent bundle of a complex manifold
        2. 2.2.2 The Frobenius theorem
        3. 2.2.3 The Newlander–Nirenberg theorem
      3. 2.3 The operators and
        1. 2.3.1 Definition
        2. 2.3.2 Local exactness
        3. 2.3.3 Dolbeault complex of a holomorphic bundle
      4. 2.4 Examples of complex manifolds
      5. Exercises
    3. 3. Kähler Metrics
      1. 3.1 Definition and basic properties
        1. 3.1.1 Hermitian geometry
        2. 3.1.2 Hermitian and Kähler metrics
        3. 3.1.3 Basic properties
      2. 3.2 Characterisations of Kähler metrics
        1. 3.2.1 Background on connections
        2. 3.2.2 Kähler metrics and connections
      3. 3.3 Examples of Kähler manifolds
        1. 3.3.1 Chern form of line bundles
        2. 3.3.2 Fubini–Study metric
        3. 3.3.3 Blowups
      4. Exercises
    4. 4. Sheaves and Cohomology
      1. 4.1 Sheaves
        1. 4.1.1 Definitions, examples
        2. 4.1.2 Stalks, kernels, images
        3. 4.1.3 Resolutions
      2. 4.2 Functors and derived functors
        1. 4.2.1 Abelian categories
        2. 4.2.2 Injective resolutions
        3. 4.2.3 Derived functors
      3. 4.3 Sheaf cohomology
        1. 4.3.1 Acyclic resolutions
        2. 4.3.2 The de Rham theorems
        3. 4.3.3 Interpretations of the group H1
      4. Exercises
  8. II: The Hodge Decomposition
    1. 5. Harmonic Forms and Cohomology
      1. 5.1 Laplacians
        1. 5.1.1 The L2 metric
        2. 5.1.2 Formal adjoint operators
        3. 5.1.3 Adjoints of the operators ∂
        4. 5.1.4 Laplacians
      2. 5.2 Elliptic differential operators
        1. 5.2.1 Symbols of differential operators
        2. 5.2.2 Symbol of the Laplacian
        3. 5.2.3 The fundamental theorem
      3. 5.3 Applications
        1. 5.3.1 Cohomology and harmonic forms
        2. 5.3.2 Duality theorems
      4. Exercises
    2. 6. The Case of Kähler Manifolds
      1. 6.1 The Hodge decomposition
        1. 6.1.1 Kähler identities
        2. 6.1.2 Comparison of the Laplacians
        3. 6.1.3 Other applications
      2. 6.2 Lefschetz decomposition
        1. 6.2.1 Commutators
        2. 6.2.2 Lefschetz decomposition on forms
        3. 6.2.3 Lefschetz decomposition on the cohomology
      3. 6.3 The Hodge index theorem
        1. 6.3.1 Other Hermitian identities
        2. 6.3.2 The Hodge index theorem
      4. Exercises
    3. 7. Hodge Structures and Polarisations
      1. 7.1 Definitions, basic properties
        1. 7.1.1 Hodge structure
        2. 7.1.2 Polarisation
        3. 7.1.3 Polarised varieties
      2. 7.2 Examples
        1. 7.2.1 Projective space
        2. 7.2.2 Hodge structures of weight 1 and abelian varieties
        3. 7.2.3 Hodge structures of weight 2
      3. 7.3 Functoriality
        1. 7.3.1 Morphisms of Hodge structures
        2. 7.3.2 The pullback and the Gysin morphism
        3. 7.3.3 Hodge structure of a blowup
      4. Exercises
    4. 8. Holomorphic de Rham Complexes and Spectral Sequences
      1. 8.1 Hypercohomology
        1. 8.1.1 Resolutions of complexes
        2. 8.1.2 Derived functors
        3. 8.1.3 Composed functors
      2. 8.2 Holomorphic de Rham complexes
        1. 8.2.1 Holomorphic de Rham resolutions
        2. 8.2.2 The logarithmic case
        3. 8.2.3 Cohomology of the logarithmic complex
      3. 8.3 Filtrations and spectral sequences
        1. 8.3.1 Filtered complexes
        2. 8.3.2 Spectral sequences
        3. 8.3.3 The Frölicher spectral sequence
      4. 8.4 Hodge theory of open manifolds
        1. 8.4.1 Filtrations on the logarithmic complex
        2. 8.4.2 First terms of the spectral sequence
        3. 8.4.3 Deligne’s theorem
      5. Exercises
  9. III: Variations of Hodge Structure
    1. 9. Families and Deformations
      1. 9.1 Families of manifolds
        1. 9.1.1 Trivialisations
        2. 9.1.2 The Kodaira–Spencer map
      2. 9.2 The Gauss–Manin connection
        1. 9.2.1 Local systems and flat connections
        2. 9.2.2 The Cartan–Lie formula
      3. 9.3 The Kähler case
        1. 9.3.1 Semicontinuity theorems
        2. 9.3.2 The Hodge numbers are constant
        3. 9.3.3 Stability of Kähler manifolds
    2. 10. Variations of Hodge Structure
      1. 10.1 Period domain and period map
        1. 10.1.1 Grassmannians
        2. 10.1.2 The period map
        3. 10.1.3 The period domain
      2. 10.2 Variations of Hodge structure
        1. 10.2.1 Hodge bundles
        2. 10.2.2 Transversality
        3. 10.2.3 Computation of the differential
      3. 10.3 Applications
        1. 10.3.1 Curves
        2. 10.3.2 Calabi–Yau manifolds
      4. Exercises
  10. IV: Cycles and Cycle Classes
    1. 11. Hodge Classes
      1. 11.1 Cycle class
        1. 11.1.1 Analytic subsets
        2. 11.1.2 Cohomology class
        3. 11.1.3 The Kähler case
        4. 11.1.4 Other approaches
      2. 11.2 Chern classes
        1. 11.2.1 Construction
        2. 11.2.2 The Kähler case
      3. 11.3 Hodge classes
        1. 11.3.1 Definitions and examples
        2. 11.3.2 The Hodge conjecture
        3. 11.3.3 Correspondences
      4. Exercises
    2. 12. Deligne–Beilinson Cohomology and the Abel–Jacobi Map
      1. 12.1 The Abel–Jacobi map
        1. 12.1.1 Intermediate Jacobians
        2. 12.1.2 The Abel–Jacobi map
        3. 12.1.3 Picard and Albanese varieties
      2. 12.2 Properties
        1. 12.2.1 Correspondences
        2. 12.2.2 Some results
      3. 12.3 Deligne cohomology
        1. 12.3.1 The Deligne complex
        2. 12.3.2 Differential characters
        3. 12.3.3 Cycle class
      4. Exercises
  11. Bibliography
  12. Index