  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 obtain that any dominant rational map from a very general
abelian $g$-fold to $\mathbb{P}^g$ has degree at least $(3\dim A+1)/2$ for
$g\geq 3$. This improves results of Alzati and the last-named author in the
case of a very general abelian variety.
