Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups

Details Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups

1999-10-21

arxiv.org/abs/math/9910113v14

Mathematics

Algebraic Geometry

33 pages, the bijectivity of the Albanese map is treated in (0.4), and a weak Lefschetz-type theorem for (higher) Chow groups

Scientific paper

We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V. Srinivas in the surface case. We then show that the Abel-Jacobi map for codimension 2 cycles and the Albanese map are bijective if a general hyperplane section is a surface for which Bloch's conjecture is proved. In certain cases we verify Nori's conjecture on the Griffiths group. We also prove a weak Lefschetz-type theorem for (higher) Chow groups, generalize a formula for the Abel-Jacobi map of higher cycles due to Beilinson and Levine to the smooth non proper case, and give a sufficient condition for the nonvanishing of the transcendental part of the image by the Abel-Jacobi map of a higher cycle on an elliptic surface, together with some examples.

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