Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map

TL;DR

This paper explores the geometric and algebraic properties of abelian varieties, focusing on their relation to the Abel-Jacobi map, the integral Hodge conjecture, and the existence of universal cycles, revealing deep connections in algebraic geometry.

Contribution

It establishes new links between the properties of abelian varieties and the integral Hodge conjecture, as well as universal cycle existence problems.

Findings

Relation between abelian varieties being summands and the integral Hodge conjecture.

Connection between split property and universal zero-cycle existence.

Analogous results for universal codimension 2 cycles on cubic threefolds.

Abstract

We discuss two properties of an abelian variety, namely, being a direct summand in a product of Jacobians and the weaker property of being "split". We relate the first property to the integral Hodge conjecture for curve classes on abelian varieties. We also relate both properties to the existence problem for universal zero-cycles on Brauer-Severi varieties over abelian varieties. A similar relation is established for the existence problem of a universal codimension 2 cycle on a cubic threefold.