Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

China Internet Development Report 2018: Blue Book of World Internet Conference In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology.

The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety - and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups - as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology.

An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Book Details

• Author: Claire Voisin
• Publisher: Princeton University Press
• eBook ISBN: 9781400850532, 1400850533
• Print ISBN: 9780691160504, 0691160503
• Other ISBNs: 0691160511, 9780691160511