An upper bound for polynomial volume growth of automorphisms of zero entropy

Fei Hu, Chen Jiang
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引用次数: 0

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.
零熵自形体的多项式体积增长上限
假设 $X$ 是一个代数封闭域上维数为 $d$ 的正射影变,而 $f$ 是 $X$ 的一个自变。假设 $f$ 在实 N\'eron--Severi 空间 $mathsf{N}^1(X)_\mathbf{R}$ 上的拉回 $f^*|_{\mathsf{N}^1(X)_\mathbf{R}}$ 是单能的,并用 $k+1$ 表示特征值 $1$ 的索引。我们将证明 $f$ 的多项式体积增长 $\mathrm{plov}(f)$ 的上界如下:\[ \mathrm{plov}(f) \le (k/2 +1)d. \] 这个不等式在某些情况下是最优的。此外,我们还证明了$k\le 2(d-1)$,将Dinh--Lin--Oguiso--Zhang对紧凑K\"ahler流形的一个结果扩展到了任意特性。把这两个不等式结合在一起,我们得到了一个最优不等式,即 \[ \mathrm{plov}(f)\le d^2, \],它肯定地回答了Cantat--Paris--Romaskevich和Lin--Oguiso--Zhang的问题。
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