Jump to ContentJump to Main Navigation
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)$
Users without a subscription are not able to see the full content.

Claire Voisin

Print publication date: 2014

Print ISBN-13: 9780691160504

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691160504.001.0001

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2022. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 16 May 2022

Decomposition of the Diagonal

Decomposition of the Diagonal

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

Claire Voisin

Princeton University Press

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

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.