An upper bound for polynomial volume growth of automorphisms of zero entropy
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918498686468096 |
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| author | Hu, Fei Jiang, Chen |
| author_facet | Hu, Fei Jiang, Chen |
| contents | 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 Néron--Severi space $\mathsf{N}^1(X)_\mathbf{R}$ is unipotent and denote the index of the eigenvalue $1$ by $k+1$. We establish the following upper bound for the polynomial volume growth $\mathrm{plov}(f)$ of $f$: \[ \mathrm{plov}(f) \le (k/2 + 1)d. \] This inequality is optimal in certain cases. Moreover, we prove that $k\le 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact Kähler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound \[ \mathrm{plov}(f) \le d^2, \] that affirmatively answers the questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15804 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An upper bound for polynomial volume growth of automorphisms of zero entropy Hu, Fei Jiang, Chen Algebraic Geometry Dynamical Systems Rings and Algebras 14J50, 16P90, 05E14, 16S38 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 Néron--Severi space $\mathsf{N}^1(X)_\mathbf{R}$ is unipotent and denote the index of the eigenvalue $1$ by $k+1$. We establish the following upper bound for the polynomial volume growth $\mathrm{plov}(f)$ of $f$: \[ \mathrm{plov}(f) \le (k/2 + 1)d. \] This inequality is optimal in certain cases. Moreover, we prove that $k\le 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact Kähler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound \[ \mathrm{plov}(f) \le d^2, \] that affirmatively answers the questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang. |
| title | An upper bound for polynomial volume growth of automorphisms of zero entropy |
| topic | Algebraic Geometry Dynamical Systems Rings and Algebras 14J50, 16P90, 05E14, 16S38 |
| url | https://arxiv.org/abs/2408.15804 |