Hodge Theory and Complex Algebraic Geometry I: Volume 1

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure written for students.

Claire Voisin (Author), Leila Schneps (Translated by)

9780521802604, Cambridge University Press

Hardback, published 5 December 2002

336 pages, 30 exercises
23.6 x 15.8 x 2.3 cm, 0.57 kg

Prize Winner Cambridge University Press congratulates Claire Voisin, winner of the 2007 Ruth Lyttle Satter Prize in Mathematics!

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.

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.

Subject Areas: Algebraic topology [PBPD], Algebraic geometry [PBMW]