Hodge Theory and Complex Algebraic Geometry II: Volume 2: (Series Number 77 Cambridge Studies in Advanced Mathematics)

The 2003 second volume of this account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard–Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether–Lefschetz theorems, the generic triviality of the Abel–Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry.

Table of Contents:
Introduction. Part I. The Topology of Algebraic Varieties: 1. The Lefschetz theorem on hyperplane sections; 2. Lefschetz pencils; 3. Monodromy; 4. The Leray spectral sequence; Part II. Variations of Hodge Structure: 5. Transversality and applications; 6. Hodge filtration of hypersurfaces; 7. Normal functions and infinitesimal invariants; 8. Nori's work; Part III. Algebraic Cycles: 9. Chow groups; 10. Mumford' theorem and its generalisations; 11. The Bloch conjecture and its generalisations; References; Index.

About the Author :
Claire Voisin is a Professor at the Institut des Hautes Études Scientifiques, France

Review :
'All together, the author has maintained her masterly style also throughout this second, much more advanced volume, just as expected. The entire two-volume text is highly instructive, inspiring, reader-friendly and generally outstanding. Without any doubt, these two volumes must be seen as an indispensible standard text on transcendental algebraic geometry for advanced students, teachers, and also researchers in this contemporary field of mathematics. The author provides, simultaneously and in a unique manner, both a complete didactic exposition and an up-to-date presentation of the subject, which is still a rather exceptional feature in the textbook literature.' Zentralblatt MATH '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 '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 '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!