Bloch’s Conjecture, Deligne Cohomology and Higher Chow Groups

This conjecture was proved in [9] if X is not of general type, but the general case still remains open. It was suggested in [36] that condition (a) in the case pg(X) = 0 would be related closely to the following conditions: (b) ind limUH 3 D(U,Q(2)) = 0, where U runs over nonempty open subvarieties of X . (c) The cycle map CH(U, 1)Q → H D(U,Q(2)) is surjective for any open subvarieties U of X . Here CH(U, n) is Bloch’s higher Chow group, and H D(U,Q(k)) is Q-Deligne cohomology. Note that condition (a) is equivalent to the injectivity of the Albanese map tensored with Q due to A. Roitman [33]. The equivalence of (a) and (b) has been proved by L. Barbieri-Viale and V. Srinivas [1] showing the exact sequence (see also [22], [34]):