An infinitesimal approach to the study of cycles on abelian varieties

This paper is a work in progress on Bloch’s conjecture asserting the vanishing of the Pontryagin product of a [Formula: see text] codimensional cycle on an abelian variety by [Formula: see text] zero cycles of degree zero. We prove an infinitesimal version of the conjecture and we discuss in particular, the case of [Formula: see text]-dimensional cycles.

BLOCH-TYPE CONJECTURES AND AN EXAMPLE A THREE-FOLD OF GENERAL TYPE

The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups". Voisin's method [19] (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A three-fold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.

Zero Cycles on Singular surfaces

Let X be a reduced and projective singular surface over ℂ and let → X be a resolution of singularities of X. We show that CH2(X) ≅ CH2() if and only if for i = 0, 1. This verifies a conjecture of Srinivas.We also verify Bloch's conjecture for singular surfaces assuming it holds for smooth surfaces. As a byproduct, we give an application to projective modules on certain singular affine surfaces.