  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.
