Polynomial log-volume growth and the GK-dimensions of twisted homogeneous coordinate rings

Abstract

Let $f$ be a zero entropy automorphism of a compact Kähler manifold $X$. We study the polynomial log-volume growth $\mathrm{Plov}(f)$ of $f$ in light of the dynamical filtrations introduced in our previous work with T.-C. Dinh. We obtain new upper bounds and lower bounds of $\mathrm{Plov}(f)$. As a corollary, we completely determine $\mathrm{Plov}(f)$ when $\dim X=3$, extending a result of Artin–Van den Bergh for surfaces. When $X$ is projective, $\mathrm{Plov}(f)+1$ coincides with the Gelfand–Kirillov dimensions $\mathrm{GKdim}(X,f)$ of the twisted homogeneous coordinate rings associated to $(X,f)$. Reformulating these results for $\mathrm{GKdim}(X,f)$, we improve Keeler’s bounds of $\mathrm{GKdim}(X,f)$ and provide effective upper bounds of $\mathrm{GKdim}(X,f)$ which only depend on $\dim X$.