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.

'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.’

Source: 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.‘

Source: 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.'

Source: Mathematical Reviews

'The book provides a very satisfying exposition of all the methods of studying algebraic cycles that have come out of Hodge theory.'

Source: Bulletin of the London Mathematical Society

‘Mathematical rewards [await] those who invest their mathematical energies in this beautiful pair of volumes.’

Source: Bulletin of the AMS

Prize Winner

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