Chow rings, decomposition of the diagonal, and the topology of families / Claire Voisin.

Includes bibliographical references (pages 155-162) and index.

Preface vii
1 Introduction 1
1.1 Decomposition of the diagonal and spread 3
1.2 The generalized Bloch conjecture 7
1.3 Decomposition of the small diagonal and application to the topology of families 9
1.4 Integral coefficients and birational invariants 11
1.5 Organization of the text 13

2 Review of Hodge theory and algebraic cycles 15
2.1 Chow groups 15
2.2 Hodge structures 24

3 Decomposition of the diagonal 36
3.1 A general principle 36
3.2 Varieties with small Chow groups 44

4 Chow groups of large coniveau complete intersections 55
4.1 Hodge coniveau of complete intersections 55
4.2 Coniveau 2 complete intersections 64
4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67
4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86

5 On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88
5.1 Tautological ring of a K3 surface 88
5.2 A decomposition of the small diagonal 96
5.3 Deligne's decomposition theorem for families of K3 surfaces 106

6 Integral coefficients 123
6.1 Integral Hodge classes and birational invariants 123
6.2 Rationally connected varieties and the rationality problem 127
6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139

Bibliography 155
Index 163

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. This title delves into arguments originating in Nori's work that have been further developed by others.