Hodge Theory and Complex Algebraic Geometry I Volume 1 [Hardcover]

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure written for students.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 and culminates with the Hodge decomposition theorem. 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. The text is complemented by exercises which provide useful results in complex algebraic geometry.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 and culminates with the Hodge decomposition theorem. 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. The text is complemented by exercises which provide useful results in complex algebraic geometry.This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with 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.Introduction; Part I. Preliminaries: 1. Holomorphic functions of many variables; 2. Complex manifolds; 3. K?hler metrics; 4. Sheaves and cohomology; Part II. The Hodge Decomposition: 5. Harmonic forms and cohomology; 6. The case of K?hler manifolds; 7. Hodge structures and polarisations; 8. Holomol“'