RESUMEN

Let X be a projective algebraic manifold and let CH^r(X) be the chow group of algebraic cycles of codimension r on X, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration  on CH^r(X) due to the second author, we construct a space of arithmetic Hodge theoretic invariants  and corresponding map , and determine conditions on X for which the kernel and image of  are “uncountably large”.