Essential dimensions of polarized endomorphisms of abelian varieties

Abstract

Let $f$ be a polarized endomorphism of an abelian variety $A$. Kollár and Zhuang asked whether the essential dimension $\mathrm{ed}(f)$ equals $\dim (A)$. We provide counterexamples to this question. Instead, we prove that, under the hypothesis that every subtorus of $A$ is $f$-preperiodic up to translation (a condition arising from the dynamical Manin--Mumford conjecture), we have $\mathrm{ed}(f^s)=\dim (A)$ for some integer $s>0$. Our examples also show the necessity of both the hypothesis and iteration. We also give an affirmative answer to Kollár and Zhuang's original question when $A$ is a simple abelian surface and $f$ is not $2$-polarized.

34 pages