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A Filtration on the Chow Groups of a Complex Projective Variety

Published online by Cambridge University Press:  04 December 2007

James D. Lewis
James D. Lewis
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. E-mail: Lewisjd@gpu.srv.ualberta.ca
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Abstract

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Let X/C be a projective algebraic manifold, and further let CHk(X)Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {F[ell ]}[ell ] [ges ] 0 on CHk(X)Q of ‘Bloch-Beilinson’ type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that Fk+1 = 0.

Keywords

Chow groupAbel–Jacobi mapregulatorDeligne cohomologyHodge structure
Type
Research Article
Information
Compositio Mathematica , Volume 128 , Issue 3 , September 2001 , pp. 299 - 322
Copyright
© 2001 Kluwer Academic Publishers