On the realizability of arithmetic degrees of morphisms

TL;DR

This paper explores which eigenvalues of surjective endomorphisms can be realized as arithmetic degrees, revealing differences between abelian and toric varieties and applying the minimal model program to this problem.

Contribution

It demonstrates that not all eigenvalues of endomorphisms on abelian varieties are arithmetic degrees, but all eigenvalues on toric varieties are, and discusses the application of the minimal model program to this question.

Findings

Not all eigenvalues of endomorphisms on abelian varieties are arithmetic degrees.

All eigenvalues of surjective endomorphisms on toric varieties are arithmetic degrees.

The minimal model program can be used to study realizability of eigenvalues as arithmetic degrees.

Abstract

The Kawaguchi-Silverman conjecture relates two different invariants of a surjective endomorphism, the dynamical and arithmetic degrees. As the Kawaguchi-Silverman conjecture is only meaningful when a morphism has a Zariski dense orbit, it has no content for varieties with positive Kodaira dimension. A generalization of the Kawaguchi-Silverman conjecture which is meaningful in positive Kodaira dimension is the so called sAND conjecture, which involves the set of "small" arithmetic degrees. Kawaguchi and Silverman showed that a small arithmetic degree is the modulus of an eigenvalue of $f^*\colon N^1(X)\rightarrow N^1(X)$. In this article we investigate which possible eigenvalues arise as an arithmetic degree. We show that surjective endomorphisms of abelian varieties may have eigenvalues which are not arithmetic degrees. Conversely, we show that every eigenvalue of a surjective endomorphism of a toric variety is an arithmetic degree using the minimal model program. Finally, we investigate how the minimal model program may be applied to study this realizability question for varieties that admit an int-amplified endomorphism.