On the Injectivity of Cycle Maps

Abstract

Let L denote also the induced filtration on CH(X)Q. In [17, II] we showed that the cycle map (0.1) is surjective if the Hodge conjecture is true for any smooth projective varieties. In this case, the Gr|cl are bijective, and (0.2) holds with the left hand side replaced by GrlCH(Z)Q, and f, / by 2p, p (but L may be non separated). The existence of such a filtration was suggested by Bloch [4]. The injectivity of (0.1) is equivalent to the separatedness of the filtration L on CE(X)Q, and would imply Bloch's conjecture [4]. The bijectivity of (0.1) is related with a problem that MM (Spec C, Q), the category of (still conjectural) mixed motives (cf. [1]) with base field C and Q-coefficients, might be close to the category of mixed Hodge structures of geometric origin