  We relate the torsion part of the Abel-Jacobi kernel in the Griffiths group
of 1-cycles to a birational invariant analogous to the degree 4 unramified
cohomology and an invariant associated to the generalized Hodge conjecture in
degree 2dim(X)-3. We also describe in terms of H-cohomology the Griffiths group
of 1-cycles and the group of torsion cycles algebraically equivalent to zero of
arbitrary dimension.
