On the virtual invariants of zero entropy groups of compact Kähler manifolds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909753114886144 |
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| author | Dinh, Tien-Cuong Lin, Hsueh-Yung Oguiso, Keiji Zhang, De-Qi |
| author_facet | Dinh, Tien-Cuong Lin, Hsueh-Yung Oguiso, Keiji Zhang, De-Qi |
| contents | Let $X$ be a compact Kähler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -κ(X)$, where $κ(X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{vir}}(G) \le \dim X -κ(X)$ for the virtual nilpotency class $c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04980 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the virtual invariants of zero entropy groups of compact Kähler manifolds Dinh, Tien-Cuong Lin, Hsueh-Yung Oguiso, Keiji Zhang, De-Qi Algebraic Geometry Complex Variables Dynamical Systems Let $X$ be a compact Kähler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -κ(X)$, where $κ(X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{vir}}(G) \le \dim X -κ(X)$ for the virtual nilpotency class $c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds. |
| title | On the virtual invariants of zero entropy groups of compact Kähler manifolds |
| topic | Algebraic Geometry Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2310.04980 |