Let $X$ be a smooth complex projective variety of dimension $d$ and $f$ an automorphism of $X$.

Suppose that the pullback $f^{*}$ of $f$ on the real Néron–Severi space $N^1(X)_R$ is unipotent and denote the index of the eigenvalue $1$ by $k+1$.

We prove an upper bound for the polynomial volume growth of $f$ (denoted by $\mathrm{plov}(f)$), or equivalently, for the Gelfand–Kirillov dimension of the twisted homogeneous coordinate ring associated with $(X, f)$, as follows:

$\mathrm{plov}(f)\leq (k/2 + 1)d.$

Combining with the inequality $k \leq 2(d-1)$ due to Dinh–Lin–Oguiso–Zhang, we obtain an optimal inequality that

$\mathrm{plov}(f)\leq d^2$,

which affirmatively answers questions of Cantat–Paris-Romaskevich and Lin–Oguiso–Zhang.

This is joint work with Chen Jiang.