# Chow rings, decomposition of the diagonal, and the topology of families

The present book originates from lectures that Claire Voisin delivered on topics related to algebraic cycles on complex algebraic varieties. The volume is intended for both students and researchers, and presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures.

The book starts with a nice and comprehensive Introduction to the topics treated on the coming chapters. Chapter 2 gives a review of the theory of Chow groups and Chow motives, and the theory of Hodge structures and mixed Hodge structures, where the Hodge Conjecture and generalized Hodge Conjecture are introduced and put into context. In Chapter 3, one of the central objects of the book is studied, the Chow class of the diagonal of $X\times X$ for a variety $X$. A result on the decomposition of the diagonal, depending on the size of Chow groups, is reviewed. The generalized Bloch conjecture is a converse statement, saying that if the trascendental cohomology of $X$ is supported on a closed algebraic set of codimension $\geq c$, then for any $i\leq c-1$, the map $CH_i(X)_{\mathbb Q}\to H^{2n-2i}(X,{\mathbb Q})$ is injective. This conjecture is a central point for the results treated in the book. Chapter 4 is devoted to the study of Chow groups for complete intersections, and the equivalence of Bloch and Hodge conjectures for general complete intersections. In Chapter 5, the author studies the small diagonal in $X\times X \times X$, which is the appropriate object to understand the ring structure on the Chow and cohomology groups. In this regard, a very particular property satisfied by the Chow ring of K3 surfaces is shown, which leads to conjectures for hyper-Kähler manifolds. The final Chapter is devoted to analysing Chow groups with integer coefficients. It is known that the Hodge Conjecture fails for integer coefficients, a fact that it is reviewed by extracting torsion invariants associated to complex cobordism groups.

This is a dense and very thorough book that reports some of the exciting discoveries that Claire Voisin has made in the study of algebraic cycles. There is a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field.

**Submitted by Vicente Munoz |

**4 / May / 2017