Canonical heights for abelian group actions of maximal dynamical rank
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915171797041152 |
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| author | Hu, Fei Zhong, Guolei |
| author_facet | Hu, Fei Zhong, Guolei |
| contents | Let $X$ be a smooth projective variety of dimension $n\geq 2$ and $G\cong\mathbf{Z}^{n-1}$ a free abelian group of automorphisms of $X$ over $\overline{\mathbf{Q}}$. Suppose that $G$ is of positive entropy. We construct a canonical height function $\widehat{h}_G$ associated with $G$, corresponding to a nef and big $\mathbf{R}$-divisor, satisfying the Northcott property. By characterizing its null locus, we prove the Kawaguchi--Silverman conjecture for each element of $G$. As another application, we determine the height counting function for non-periodic points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_17759 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Canonical heights for abelian group actions of maximal dynamical rank Hu, Fei Zhong, Guolei Number Theory Algebraic Geometry Dynamical Systems 11G50, 14J50, 37P15, 37P30 Let $X$ be a smooth projective variety of dimension $n\geq 2$ and $G\cong\mathbf{Z}^{n-1}$ a free abelian group of automorphisms of $X$ over $\overline{\mathbf{Q}}$. Suppose that $G$ is of positive entropy. We construct a canonical height function $\widehat{h}_G$ associated with $G$, corresponding to a nef and big $\mathbf{R}$-divisor, satisfying the Northcott property. By characterizing its null locus, we prove the Kawaguchi--Silverman conjecture for each element of $G$. As another application, we determine the height counting function for non-periodic points. |
| title | Canonical heights for abelian group actions of maximal dynamical rank |
| topic | Number Theory Algebraic Geometry Dynamical Systems 11G50, 14J50, 37P15, 37P30 |
| url | https://arxiv.org/abs/2311.17759 |