Chow Rings Decomposition of the Diagonal and the Topology of Families Annals of Mathematics Studies

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. 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-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 -- 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. Claire Voisin. Includes Bibliographical References And Index. ""4 Chow groups of large coniveau complete intersections""""4.1 Hodge coniveau of complete intersections""; ""4.2 Coniveau 2 complete intersections""; ""4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections""; ""4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces""; ""5 On the Chow ring of K3 surfaces and hyper-KÃ?hler manifolds""; ""5.1 Tautological ring of a K3 surface""; ""5.2 A decomposition of the small diagonal""; ""5.3 Deligne's decomposition theorem for families of K3 surfaces""; ""6 Integral coefficients ""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""1 Introduction""; ""1.1 Decomposition of the diagonal and spread""; ""1.2 The generalized Bloch conjecture""; ""1.3 Decomposition of the small diagonal and application to the topology of families""; ""1.4 Integral coefficients and birational invariants""; ""1.5 Organization of the text""; ""2 Review of Hodge theory and algebraic cycles""; ""2.1 Chow groups""; ""2.2 Hodge structures""; ""3 Decomposition of the diagonal""; ""3.1 A general principle""; ""3.2 Varieties with small Chow groups "The rise of China could be the most important political development of the twenty-first century. What will China look like in the future? What should it look like? And what will China's rise mean for the rest of world? This book, written by China's most influential foreign policy thinker, sets out a vision for the coming decades from China's point of view." -- Publisher's web aite ""6.1 Integral Hodge classes and birational invariants""""6.2 Rationally connected varieties and the rationality problem""; ""6.3 Integral decomposition of the diagonal and the structure of the Abelâ€?Jacobi map""; ""Bibliography""; ""Index