  We explicitly describe cycle-class maps c_H from motivic cohomology to
absolute Hodge cohomology, for smooth quasi-projective and (some) proper
singular varieties, and compute special cases of the latter. For smooth
projective varieties, we also study Hodge-theoretically defined "higher
Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial
indecomposable higher Chow cycles in the regulator kernel.
