Bloch-type conjectures and an example of a threefold 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 (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A threefold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.