Kawaguchi–Silverman conjecture on automorphisms of projective threefolds

Abstract

Under the framework of dynamics on normal projective varieties by Kawamata, Nakayama and Zhang, and Hu and Li, 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 cases: (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 $q(X)\ge \mathrm{\text{dim}}X-1$. Finally, we discuss Kawaguchi–Silverman conjecture on normal projective varieties with Picard number two.