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## 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.

1. Cover
2. Half Title
3. Title Page
5. Contents
6. 0. Introduction
7. I: Preliminaries
1. 1. Holomorphic Functions of Many Variables
1. 1.1 Holomorphic functions of one variable
2. 1.2 Holomorphic functions of several variables
3. 1.3 The equation ∂g/∂z = f
4. Exercises
2. 2. Complex Manifolds
1. 2.1 Manifolds and vector bundles
2. 2.2 Integrability of almost complex structures
3. 2.3 The operators and
4. 2.4 Examples of complex manifolds
5. Exercises
3. 3. Kähler Metrics
1. 3.1 Definition and basic properties
2. 3.2 Characterisations of Kähler metrics
3. 3.3 Examples of Kähler manifolds
4. Exercises
4. 4. Sheaves and Cohomology
1. 4.1 Sheaves
2. 4.2 Functors and derived functors
3. 4.3 Sheaf cohomology
4. Exercises
8. II: The Hodge Decomposition
1. 5. Harmonic Forms and Cohomology
1. 5.1 Laplacians
2. 5.2 Elliptic differential operators
3. 5.3 Applications
4. Exercises
2. 6. The Case of Kähler Manifolds
1. 6.1 The Hodge decomposition
2. 6.2 Lefschetz decomposition
3. 6.3 The Hodge index theorem
4. Exercises
3. 7. Hodge Structures and Polarisations
1. 7.1 Definitions, basic properties
2. 7.2 Examples
3. 7.3 Functoriality
4. Exercises
4. 8. Holomorphic de Rham Complexes and Spectral Sequences
1. 8.1 Hypercohomology
2. 8.2 Holomorphic de Rham complexes
3. 8.3 Filtrations and spectral sequences
4. 8.4 Hodge theory of open manifolds
5. Exercises
9. III: Variations of Hodge Structure
1. 9. Families and Deformations
1. 9.1 Families of manifolds
2. 9.2 The Gauss–Manin connection
3. 9.3 The Kähler case
2. 10. Variations of Hodge Structure
1. 10.1 Period domain and period map
2. 10.2 Variations of Hodge structure
3. 10.3 Applications
4. Exercises
10. IV: Cycles and Cycle Classes
1. 11. Hodge Classes
1. 11.1 Cycle class
2. 11.2 Chern classes
3. 11.3 Hodge classes
4. Exercises
2. 12. Deligne–Beilinson Cohomology and the Abel–Jacobi Map
1. 12.1 The Abel–Jacobi map
2. 12.2 Properties
3. 12.3 Deligne cohomology
4. Exercises
11. Bibliography
12. Index