Some results on Green ’ s higher Abel - Jacobi map

A filtration satisfying this last property has been constructed by Saito [11], but it is not shown that the filtration terminates, that is F p+1CH(X)Q = 0. A definition has also been proposed by J. P. Murre ([8]), under the assumption that a strong Künneth decomposition of the diagonal exists, but it is not proved to satisfy condition ii) above. In fact proving the existence of such a filtration would solve in particular Bloch’s conjecture on zero-cycles of surfaces [1]. In any case, the first steps of the filtration are easy to understand, at least for zero-cycles. Namely one should have F CH0(X) = CH0(X)alb = Ker alb, where alb : CH0(X)hom → Alb(X) is the Albanese map. More generally for