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Annals of Mathematics Studies #187: Chow Rings, Decomposition of the Diagonal, and the Topology of Families


Annals of Mathematics Studies #187: Chow Rings, Decomposition of the Diagonal, and the Topology of Families Cover


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Publisher Comments:

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 Noris work that have been further developed by others.

About the Author

Claire Voisin has been a senior researcher at Frances National Center for Scientific Research since 1986.

Table of Contents

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-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

Bibliography 155

Index 163

Product Details

Voisin, Claire
Princeton University Press
Mathematics-Algebraic Geometry
Annals of Mathematics Studies
Publication Date:
9 x 6 in

Related Subjects

Science and Mathematics » Biology » Zoology » Ornithology
Science and Mathematics » Mathematics » Geometry » Algebraic Geometry
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Science and Mathematics » Mathematics » Reference
Science and Mathematics » Mathematics » Topology

Annals of Mathematics Studies #187: Chow Rings, Decomposition of the Diagonal, and the Topology of Families New Hardcover
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Product details 176 pages Princeton University Press - English 9780691160504 Reviews:
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