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 prove an upper bound for the polynomial volume growth $\mathrm{plov}(f)$ of $f$ as follows: [ \mathrm{plov}(f) \le (k/2 + 1)d. ] This inequality is optimal in certain cases. Furthermore, we show that $k\le 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact K\"ahler manifolds to arbitrary characteristic. Combining these two inequalities together, we obtain an optimal inequality that [ \mathrm{plov}(f) \le d^2, ] which affirmatively answers questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang.