Hodge Theory and Complex Algebraic Geometry I

by Claire Voisin , Translated by Leila Schneps

• ISBN13: 9780521718011
• ISBN10: 0521718015
• Edition: 1st
• Format: Paperback
• Copyright: 2008-02-04
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Introduction

Preliminaries

Holomorphic functions of many variables

Complex manifolds

K+ñhler metrics

Sheaves and cohomology

The Hodge Decomposition

Harmonic forms and cohomology

The case of K+ñhler manifolds

Hodge structures and polarisations

Holomorphic de Rham complexes

Variations of Hodge Structure

Families and deformations

Variations of Hodge structure

Cycles and Cycle Classes

Hodge classes

The AbelG++Jacobi map

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.