A lower bound for polynomial volume growth of automorphisms of zero entropy

Abstract

Let $X$ be a normal projective variety of dimension $d$, and let $f$ be a zero-entropy automorphism of $X$. Denote by $k$ the first-degree growth rate of $f$, so that $\deg_1(f^n) \asymp n^{k}$. We prove the sharp lower bound for the polynomial volume growth $\mathrm{plov}(f)$ of $f$: \[ \mathrm{plov}(f) \ge d+\frac{k(k+2)}{4}, \] equivalently giving a sharp lower bound on the Gelfand--Kirillov dimension of the associated twisted homogeneous coordinate ring. This improves previous lower bounds of Keeler and of Lin--Oguiso--Zhang. In the proof, we introduce the notion of dynamical intersection polynomials and give a new characterization of $\mathrm{plov}(f)$ in terms of non-vanishing of intersection numbers. We also establish a gap principle for polynomial volume growth: for every fixed dimension $d\ge 4$, either $\mathrm{plov}(f)=d^2$, or $\mathrm{plov}(f)\le d(d-2) + 2\lfloor d/4 \rfloor$. This reveals a new rigidity phenomenon for zero-entropy automorphisms. As an application, in dimension $4$ we determine all possible values of $\mathrm{plov}$, thereby extending the results of Artin--Van den Bergh for surfaces and Lin--Oguiso--Zhang for threefolds.