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Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)$
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Claire Voisin

Print publication date: 2014

Print ISBN-13: 9780691160504

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691160504.001.0001

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Decomposition of the Diagonal

Decomposition of the Diagonal

Chapter:
(p.36) Chapter Three Decomposition of the Diagonal
Source:
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)
Author(s):

Claire Voisin

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691160504.003.0003

This chapter explains the method initiated by Bloch and Srinivas, which leads to statements of the following: if a smooth projective variety has trivial Chow groups of k-cycles homologous to 0 for kc1, then its transcendental cohomology has geometric coniveau ≤ c. This result is a vast generalization of Mumford's theorem. A major open problem is the converse of this result. It turns out that statements of this kind are a consequence of a general spreading principle for rational equivalence. Consider a smooth projective family XB and a cycle ZB, everything defined over C; then, if at the very general point bB, the restricted cycle Z𝒳bX𝒳b is rationally equivalent to 0, there exist a dense Zariski open set UB and an integer N such that NZsubscript U is rationally equivalent to 0 on Xsubscript U.

Keywords:   diagonal, transcendental cohomology, Mumford's theorem, rational equivalence, dense Zariski open set, smooth projective varieties

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