On a Conjecture of Voisin on the Gonality of Very General Abelian Varieties.

We adapt a method of Voisin to powers of abelian varieties in order to study orbits for rational equivalence of zero-cycles on very general abelian varieties. We deduce that a very general abelian variety of dimension at least $2k-2$ has gonality at least $k+1$. 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 Sommese bound on the degree of irrationality of abelian varieties. In the appendix we present some new identities in the Chow group of zero-cycles of abelian varieties.