Essential dimensions of polarized endomorphisms of abelian varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908694263889920 |
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| author | Luo, Yujie Oguiso, Keiji Zhang, De-Qi |
| author_facet | Luo, Yujie Oguiso, Keiji Zhang, De-Qi |
| contents | Let $f$ be a polarized endomorphism of an abelian variety $A$. Kollár and Zhuang asked whether the essential dimension $\mathrm{ed}(f)$ equals $\mathrm{dim}(A)$. We provide counterexamples to this question. Instead, we prove that, under the hypothesis that every subtorus of $A$ is $f$-preperiodic up to translation (a condition arising from the dynamical Manin--Mumford conjecture), we have $\mathrm{ed}(f^s)=\mathrm{dim}(A)$ for some integer $s>0$. Our examples also show the necessity of both the hypothesis and iteration. We also give an affirmative answer to Kollár and Zhuang's original question when $A$ is a simple abelian surface and $f$ is not $2$-polarized. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_04942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Essential dimensions of polarized endomorphisms of abelian varieties Luo, Yujie Oguiso, Keiji Zhang, De-Qi Algebraic Geometry Dynamical Systems 14K02 (Primary) 08A35 (Secondary) Let $f$ be a polarized endomorphism of an abelian variety $A$. Kollár and Zhuang asked whether the essential dimension $\mathrm{ed}(f)$ equals $\mathrm{dim}(A)$. We provide counterexamples to this question. Instead, we prove that, under the hypothesis that every subtorus of $A$ is $f$-preperiodic up to translation (a condition arising from the dynamical Manin--Mumford conjecture), we have $\mathrm{ed}(f^s)=\mathrm{dim}(A)$ for some integer $s>0$. Our examples also show the necessity of both the hypothesis and iteration. We also give an affirmative answer to Kollár and Zhuang's original question when $A$ is a simple abelian surface and $f$ is not $2$-polarized. |
| title | Essential dimensions of polarized endomorphisms of abelian varieties |
| topic | Algebraic Geometry Dynamical Systems 14K02 (Primary) 08A35 (Secondary) |
| url | https://arxiv.org/abs/2512.04942 |