Ebook: Chow Rings, Decomposition of the Diagonal, and the Topology of Families

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.

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These notes grew out of lectures I delivered in Philadelphia (the Rademacher lectures) and in Princeton (the Hermann Weyl lectures). The central objects are the diagonal of a variety X and the small diagonal in X3. Topologically, we have the Kunneth decomposition of the diagonal, which has as a consequence, for example, the Poincare{Hopf formula, but such a Kunneth decomposition does not exist in the context of Chow groups, unless the variety has trivial Chow groups. The Bloch{Srinivas principle and its generalizations provide the beginning of such a decomposition in the Chow group of X  X, under the assumption that Chow groups in small dimension are trivial (that is, parametrized by the cohomological cycle class). The study of the diagonal thus allows us to study Chow groups CH(X) of X seen additively, but not the ring structure of CH(X). The latter is governed by the small diagonal, which, seen as a correspondence between X X and X, induces the cup-product in cohomology and the intersection product on Chow groups. The second central topic of the book is the spread of cycles and rational equivalence, which appeared rst in Nori's work and which has become very important to relate Chow groups and topology in a rened way. I rst considered the small diagonal in joint work with Beauville where we proved that the small diagonal of a K3 surface has a very special Chow-theoretic decomposition. I then realized that this partially extends to some Calabi{Yau varieties, and furthermore that this decomposition, when spread up over a family, implies very special multiplicative properties of the Leray spectral sequence. Concerning the diagonal itself, I proved recently, by a spreading argument applied to a cohomological decomposition of the diagonal, that for varieties like complete intersections, admitting large families of deformations with very simple total space, the generalized Hodge conjecture predicting equality between the Hodge coniveau and the geometric coniveau is equivalent to the generalized Bloch conjecture saying that the Hodge coniveau governs the triviality of Chow groups of small dimension. This book also re ects my interest in recent years in questions involving cycles with Z-coecients rather than Q-coecients. The diagonal decomposition and, more generally, the spreading principle for rational equivalence, become wrong with Z-coecients, and this is a source of interesting torsion invariants. I have also included a discussion of the defect of the Hodge conjecture with integral coecients, as the recent proof of the Bloch{Kato conjecture gave an important new impulse to the subject.