An upper bound for polynomial volume growth of automorphisms of zero entropy

Abstract

Let $X$ be a normal projective variety of dimension $d$ over an algebraically closed field and $f$ an automorphism of $X$. Suppose that the pullback $f^*|_{\mathsf{N}^1(X)_\mathbf{R}}$ of $f$ on the real N\'eron--Severi space $\mathsf{N}^1(X)_\mathbf{R}$ is unipotent and denote the index of the eigenvalue $1$ by $k+1$. We establish the following upper bound for the polynomial volume growth $\mathrm{plov}(f)$ of $f$: \[ \mathrm{plov}(f) \le (k/2 + 1)d. \] This inequality is optimal in certain cases. Moreover, we prove that $k\le 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact K\"ahler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound \[ \mathrm{plov}(f) \le d^2, \] that affirmatively answers the questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang.