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

TL;DR

This paper proves Bloch's conjecture for certain correspondences on powers of complex abelian varieties, establishing new vanishing results and addressing questions related to K3 surfaces and Kummer varieties.

Contribution

It introduces a proof of Bloch's conjecture for generically defined correspondences and advances the understanding of the generalized Hodge conjecture for abelian varieties.

Findings

Bloch's conjecture holds for generically defined correspondences on powers of abelian varieties.

Vanishing results for symmetric and skew-symmetric cycles on abelian varieties.

Symplectic automorphisms of Kummer varieties act trivially on zero-cycles.

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.