Introduction; Part I. Preliminaries: 1. Holomorphic functions of many variables; 2. Complex manifolds; 3. Kahler metrics; 4. Sheaves and cohomology; Part II. The Hodge Decomposition: 5. Harmonic forms and cohomology; 6. The case of Kahler 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.
The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure written for students.
Claire Voisin is a Professor at the Institut des Hautes Etudes Scientifiques, France
'This introductory text to Hodge theory and Kahlerian geometry is an excellent and modern introduction to the subject, shining with comprehensiveness, strictness, clarity, rigor, thematic steadfastness of purpose, and catching enthusiasm for this fascinating field of contemporary mathematical research. This book is exceedingly instructive, inspiring, challenging and user-friendly, which makes it truly outstanding and extremely valuable for students, teachers, and researchers in complex geometry.' Zentralblatt MATH 'I would recommend anyone interested in learning about a topic in complex differential or algebraic geometry to read Voisin's volumes. She has done a remarkably good job.' Proceedings of the Edinburgh Mathematical Society '... 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 'The book provides a very satisfying exposition of all the methods of studying algebraic cycles that have come out of Hodge theory.' Bulletin of the London Mathematical Society 'Mathematical rewards [await] those who invest their mathematical energies in this beautiful pair of volumes.' Bulletin of the AMS Prize Winner Cambridge University Press congratulates Claire Voisin, winner of the 2007 Ruth Lyttle Satter Prize in Mathematics!