This book provides a comprehensive and up-to-date introduction
to Hodge theory--one of the central and most vibrant areas of
contemporary mathematics--from leading specialists on the subject.
The topics range from the basic topology of algebraic varieties to
the study of variations of mixed Hodge structure and the Hodge
theory of maps. Of particular interest is the study of algebraic
cycles, including the Hodge and Bloch-Beilinson Conjectures. Based
on lectures delivered at the 2010 Summer School on Hodge Theory at
the ICTP in Trieste, Italy, the book is intended for a broad group
of students and researchers. The exposition is as accessible as
possible and doesn't require a deep background. At the same time,
the book presents some topics at the forefront of current
research.
The book is divided between introductory and advanced lectures.
The introductory lectures address Kahler manifolds, variations of
Hodge structure, mixed Hodge structures, the Hodge theory of maps,
period domains and period mappings, algebraic cycles (up to and
including the Bloch-Beilinson conjecture) and Chow groups, sheaf
cohomology, and a new treatment of Grothendieck's algebraic de Rham
theorem. The advanced lectures address a Hodge-theoretic
perspective on Shimura varieties, the spread philosophy in the
study of algebraic cycles, absolute Hodge classes (including a new,
self-contained proof of Deligne's theorem on absolute Hodge
cycles), and variation of mixed Hodge structures.
The contributors include Patrick Brosnan, James Carlson, Eduardo
Cattani, Francois Charles, Mark Andrea de Cataldo, Fouad El Zein,
Mark L. Green, Phillip A. Griffiths, Matt Kerr, Le D?ng Trang, Luca
Migliorini, Jacob P. Murre, Christian Schnell, and Loring W.
Tu."
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