Indecomposable Higher Chow Cycles

Abstract

Let X be a projective algebraic manifold. A short survey is given on what is known about Hodge theoretic obstructions to representability of Chow groups. We begin by discussing Mumford’s famous theorem for zero cycles on surfaces of positive genus, and a variant of this theorem for a given higher Chow group. To this end, a generalization of Mumford’s theorem to higher Chow groups is given, with an example of a product of three elliptic curves, whose regulator kernel is large, even modulo decomposables.