ar X iv : 0 70 5 . 46 61 v 1 [ m at h . A G ] 3 1 M ay 2 00 7 ALGEBRAIC CYCLES AND MUMFORD - GRIFFITHS INVARIANTS

Let X be a projective algebraic manifold and let CH(X) be the Chow group of algebraic cycles of codimension r on X, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration {F }ν≥0 on CH(X)⊗Q due to the second author, we construct a space of arithmetic Hodge theoretic invariants ∇J(X) and corresponding map φ X : Gr F CH(X)⊗Q → ∇J(X), and determine conditions on X for which the kernel and image of φ X are “uncountably large”.