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

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 ${\mathrm{deg}}_1(f^n)\asymp n^k$. We prove the sharp lower bound for the polynomial volume growth $\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)$ of $f$: $$\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)\ge d+\frac{k(k+2)}{4},$$ 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 $\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)$ 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 $\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)=d^2$, or $\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)\le d(d-2)+2\lfloor d\mathrm{/}4\rfloor$. This reveals a new rigidity phenomenon for zero-entropy automorphisms. As an application, in dimension $4$ we determine all possible values of $\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}$, thereby extending the results of Artin–Van den Bergh for surfaces and Lin–Oguiso–Zhang for threefolds.