{"title":"零熵自形体的多项式体积增长上限","authors":"Fei Hu, Chen Jiang","doi":"arxiv-2408.15804","DOIUrl":null,"url":null,"abstract":"Let $X$ be a normal projective variety of dimension $d$ over an algebraically\nclosed field and $f$ an automorphism of $X$. Suppose that the pullback\n$f^*|_{\\mathsf{N}^1(X)_\\mathbf{R}}$ of $f$ on the real N\\'eron--Severi space\n$\\mathsf{N}^1(X)_\\mathbf{R}$ is unipotent and denote the index of the\neigenvalue $1$ by $k+1$. We prove an upper bound for the polynomial volume\ngrowth $\\mathrm{plov}(f)$ of $f$ as follows: \\[ \\mathrm{plov}(f) \\le (k/2 +\n1)d. \\] This inequality is optimal in certain cases. Furthermore, we show that\n$k\\le 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact\nK\\\"ahler manifolds to arbitrary characteristic. Combining these two\ninequalities together, we obtain an optimal inequality that \\[ \\mathrm{plov}(f)\n\\le d^2, \\] which affirmatively answers questions of Cantat--Paris-Romaskevich\nand Lin--Oguiso--Zhang.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An upper bound for polynomial volume growth of automorphisms of zero entropy\",\"authors\":\"Fei Hu, Chen Jiang\",\"doi\":\"arxiv-2408.15804\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $X$ be a normal projective variety of dimension $d$ over an algebraically\\nclosed field and $f$ an automorphism of $X$. Suppose that the pullback\\n$f^*|_{\\\\mathsf{N}^1(X)_\\\\mathbf{R}}$ of $f$ on the real N\\\\'eron--Severi space\\n$\\\\mathsf{N}^1(X)_\\\\mathbf{R}$ is unipotent and denote the index of the\\neigenvalue $1$ by $k+1$. We prove an upper bound for the polynomial volume\\ngrowth $\\\\mathrm{plov}(f)$ of $f$ as follows: \\\\[ \\\\mathrm{plov}(f) \\\\le (k/2 +\\n1)d. \\\\] This inequality is optimal in certain cases. Furthermore, we show that\\n$k\\\\le 2(d-1)$, extending a result of Dinh--Lin--Oguiso--Zhang for compact\\nK\\\\\\\"ahler manifolds to arbitrary characteristic. Combining these two\\ninequalities together, we obtain an optimal inequality that \\\\[ \\\\mathrm{plov}(f)\\n\\\\le d^2, \\\\] which affirmatively answers questions of Cantat--Paris-Romaskevich\\nand Lin--Oguiso--Zhang.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"75 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.15804\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.15804","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
An upper bound for polynomial volume growth of automorphisms of zero entropy
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.