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

Author: Claire Voisin
Publisher: Princeton : Princeton University Press, 2014.
Series: Annals of mathematics studies, no. 187.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

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  Read more...

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Document Type: Book
All Authors / Contributors: Claire Voisin
ISBN: 9780691160504 0691160503 9780691160511 0691160511 1400850533 9781400850532
OCLC Number: 852745692
Description: viii, 162 pages ; 24 cm.
Contents: 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.
Series Title: Annals of mathematics studies, no. 187.
Responsibility: Claire Voisin.

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"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 "[An Read more...

 
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