1999/10/21 by Morihiko Saito, Saito, Morihiko
Mathematics · #14C30 #32S35 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C30 #msc:32S35
paper · pdf · doi:10.48550/arxiv.math/9910113
33 pages, the bijectivity of the Albanese map is treated in (0.4), and a weak Lefschetz-type theorem for (higher) Chow groups is studied
arxiv created 2002/11/18 · arxiv updated 2009/11/30
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.