Synopses & Reviews
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.
Review
"This dense, fascinating book by Voisin is a report of some of the exciting discoveries she has made in the quest of the secrets of algebraic cycles."--Alberto Collino, Zentralblatt MATH
Review
"[An advanced] reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf."--C. A. M. Peters, Mathematical Reviews Clippings
About the Author
Claire Voisin has been a senior researcher at Frances National Center for Scientific Research since 1986.
Table of Contents
Preface vii
1Introduction 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
2Review of Hodge theory and algebraic cycles 15
2.1 Chow groups 15
2.2 Hodge structures 24
3Decomposition of the diagonal 36
3.1 A general principle 36
3.2 Varieties with small Chow groups 44
4Chow 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
5On the Chow ring of
K3 surfaces and hyper-Kähler 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
6Integral 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