# On a conjecture of Voisin on the gonality of very general abelian varieties

## TL;DR

This paper proves Voisin's conjecture that very general abelian varieties of certain dimensions have gonality exceeding a given bound, linking rational equivalence orbits to measures of irrationality.

## Contribution

It adapts Voisin's method to powers of abelian varieties, establishing new lower bounds on gonality and degree of irrationality for very general abelian varieties.

## Key findings

- Proves gonality exceeds k for certain abelian varieties
- Establishes new lower bounds on measures of irrationality
- Strengthens bounds on degree of irrationality of abelian varieties

## Abstract

We study orbits for rational equivalence of zero-cycles on very general abelian varieties by adapting a method of Voisin to powers of abelian varieties. We deduce that, for $k$ at least $3$, a very general abelian variety of dimension at least $2k-2$ has covering gonality greater than $k$. This settles a conjecture of Voisin. We also discuss how upper bounds for the dimension of orbits for rational equivalence can be used to provide new lower bounds on other measures of irrationality. In particular, we obtain a strengthening of the Alzati-Pirola bound on the degree of irrationality of abelian varieties.