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 l}l ≥ 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.

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Lewis, J.D. A Filtration on the Chow Groups of a Complex Projective Variety. Compositio Mathematica 128, 299–322 (2001). https://doi.org/10.1023/A:1011882030468

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  • Chow group
  • Abel–Jacobi map
  • regulator
  • Deligne cohomology
  • Hodge structure