Hodge Theory and Complex Algebraic Geometry I
Hodge Theory and Complex Algebraic Geometry I
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.
- Self-contained with full proofs, making it understandable to graduate students
- A modern treatment of the subject, now in paperback
- Exercises complement the main text, and give useful extra results
Reviews & endorsements
"...this book is going to become a very common reference in this field. ...useful for both a student trying to learn the subject as well as the researcher that can find a wealth of results in a clear and compact format. The exposition is very precise and the introduction that precedes each chapter helps the reader to focus on the main ideas in the text." Mathematical Reviews
"Mathematical rewards [await] those who invest their mathematical energies in this beautiful pair of volumes." Bulletin of the AMS
Product details
February 2008Paperback
9780521718011
334 pages
227 × 153 × 18 mm
0.53kg
30 exercises
Available
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 and spectral sequences
- 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. Deligne-Beilinson cohomology and the Abel-Jacobi map
- Bibliography
- Index.
