A Filtration on the Chow Groups of a Complex Projective Variety

Abstract

Let X/_C be a projective algebraic manifold, and further let CH^k(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 CH^k(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 F^k+1 = 0.