Kawaguchi-Silverman conjecture on automorphisms of projective threefolds

TL;DR

This paper advances understanding of the Kawaguchi-Silverman conjecture by reducing it to specific cases for automorphisms on threefolds and proves it for varieties with high irregularity, also discussing cases with Picard number two.

Contribution

It reduces the Kawaguchi-Silverman conjecture for certain threefold automorphisms to two specific cases and proves it for varieties with irregularity at least dimension minus one.

Findings

Kawaguchi-Silverman conjecture holds for automorphisms with high irregularity.

Reduction of the conjecture to primitive automorphisms of weak Calabi-Yau threefolds or rationally connected threefolds.

Discussion of the conjecture on varieties with Picard number two.

Abstract

Under the framework of dynamics on normal projective varieties by Kawamata, Nakayama and Zhang \cite{Kawamata85,Nakayama10,NZ09,NZ10,Zhang16}, Hu and the author \cite{HL21}, we may reduce Kawaguchi-Silverman conjecture for automorphisms $f$ on normal projective threefolds $X$ with either the canonical divisor $K_X$ is trivial or negative Kodaira dimension to the following two case: (i) $f$ is a primitively automorphism of a weak Calabi-Yau threefold (ii) $X$ is a rationally connected threefold. And we prove Kawaguchi-Silverman conjecture is true for automorphisms of normal projective varieties $X$ with the irregularity $q(X)\ge\dim X-1$. Finally, we discuss Kawaguchi-Silverman conjecture on normal projective varieties with Picard number two.