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Edition: XX

Copyright: 2002

Publisher: Cambridge University Press

Published: 2002

International: No

Copyright: 2002

Publisher: Cambridge University Press

Published: 2002

International: No

Edition: XX

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This is a modern introduction to Kaehlerian geometry and Hodge structure. It 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 book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. The book is is completely self-contained and can be used by students, while its content gives an up-to-date 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.

- Self-contained with full proofs, so suitable for students
- Only up-to-date treatment of subject
- Contains material never presented before in book form

Introduction

**Part I. Preliminaries: **

1. Holomorphic functions of many variables

2. Complex manifolds

3. Kähler metrics

4. Sheaves and cohomology

**Part II. The Hodge Decomposition: **

5. Harmonic forms and cohomology

6. The case of Kähler manifolds

7. Hodge structures and polarisations

8. Holomorphic de Rham complexes

**Part III. Variations of Hodge Structure: **

9. Families and deformations

10. Variations of Hodge structure

**Part IV. Cycles and Cycle Classes: **

11. Hodge classes

12. The Abel-Jacobi map

Bibliography

Index.

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Summary

This is a modern introduction to Kaehlerian geometry and Hodge structure. It 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 book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. The book is is completely self-contained and can be used by students, while its content gives an up-to-date 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.

- Self-contained with full proofs, so suitable for students
- Only up-to-date treatment of subject
- Contains material never presented before in book form

Table of Contents

Introduction

**Part I. Preliminaries: **

1. Holomorphic functions of many variables

2. Complex manifolds

3. Kähler metrics

4. Sheaves and cohomology

**Part II. The Hodge Decomposition: **

5. Harmonic forms and cohomology

6. The case of Kähler manifolds

7. Hodge structures and polarisations

8. Holomorphic de Rham complexes

**Part III. Variations of Hodge Structure: **

9. Families and deformations

10. Variations of Hodge structure

**Part IV. Cycles and Cycle Classes: **

11. Hodge classes

12. The Abel-Jacobi map

Bibliography

Index.

Publisher Info

Publisher: Cambridge University Press

Published: 2002

International: No

Published: 2002

International: No

This is a modern introduction to Kaehlerian geometry and Hodge structure. It 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 book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. The book is is completely self-contained and can be used by students, while its content gives an up-to-date 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.

- Self-contained with full proofs, so suitable for students
- Only up-to-date treatment of subject
- Contains material never presented before in book form

Introduction

**Part I. Preliminaries: **

1. Holomorphic functions of many variables

2. Complex manifolds

3. Kähler metrics

4. Sheaves and cohomology

**Part II. The Hodge Decomposition: **

5. Harmonic forms and cohomology

6. The case of Kähler manifolds

7. Hodge structures and polarisations

8. Holomorphic de Rham complexes

**Part III. Variations of Hodge Structure: **

9. Families and deformations

10. Variations of Hodge structure

**Part IV. Cycles and Cycle Classes: **

11. Hodge classes

12. The Abel-Jacobi map

Bibliography

Index.