Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for abelian varieties

Abstract: We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let $\gamma$ be a correspondence between two abelian varieties $A$ and $B$ that can be written as a linear combination of products of symmetric divisors. Assume that $A$ is isogenous to the product of an abelian variety of totally real type with the power of an abelian surface. We show that $\gamma$ satisfies the conclusion of Bloch's conjecture. A key ingredient consists in establishing a strong form of the generalized Hodge conjecture for Hodge sub-structures of the cohomology of $A$ that arise as sub-representations of the Lefschetz group of $A$. As a by-product of our method, we use a strong form of the generalized Hodge conjecture established for powers of abelian surfaces to show that every finite-order symplectic automorphism of a generalized Kummer variety acts as the identity on the zero-cycles.