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Contents

0 Introduction

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 ∂g/∂z = f

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 and

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

Exercises

3 K¨ahler Metrics

3.1 Definition and basic properties

3.1.1 Hermitian geometry

3.1.2 Hermitian and K¨ahler metrics

3.1.3 Basic properties

3.2 Characterisations of K¨ahler metrics

3.2.1 Background on connections

3.2.2 K¨ahler metrics and connections

3.3 Examples of K¨ahler 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

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 H1

Exercises

II The Hodge Decomposition

5 Harmonic Forms and Cohomology

5.1 Laplacians

5.1.1 The L2 metric

5.1.2 Formal adjoint operators

5.1.3 Adjoints of the operators ∂bar

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 K¨ahler Manifolds

6.1 The Hodge decomposition

6.1.1 K¨ahler 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

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 Fr¨olicher 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

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 K¨ahler case

9.3.1 Semicontinuity theorems

9.3.2 The Hodge numbers are constant

9.3.3 Stability of K¨ahler 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

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 K¨ahler case

11.1.4 Other approaches

11.2 Chern classes

11.2.1 Construction

11.2.2 The K¨ahler 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

Exercises

Bibliography

Index