Synopses & Reviews
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.
Review
"...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
Review
"...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
Synopsis
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.
Synopsis
A completely self-contained modern introduction to Kaehlerian geometry and Hodge structure written for students.
Synopsis
The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure written for students.
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.