Degree of irrationality of a very general abelian variety

Abstract

Abstract Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0<d\leq \dim A$. We show that if the map $A^k\to CH_0(A)$ has a $d$-dimensional fiber then $k\geq d+(\dim A+1)/2$. This extends results of the second-named author which covered the cases $d=1,2$. As a geometric application, we prove that any dominant rational map from a very general abelian $g$-fold to $\mathbb{P}^g$ has degree at least $(3g+1)/2$ for $g\geq 3$, thus improving results of Alzati and the last-named author in the case of a very general abelian variety.