On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909399530864640 |
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| author | Catanese, Fabrizio Frapporti, Davide Gleissner, Christian Liu, Wenfei Schütt, Matthias |
| author_facet | Catanese, Fabrizio Frapporti, Davide Gleissner, Christian Liu, Wenfei Schütt, Matthias |
| contents | In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $κ(S)=1$), in the initial case $ χ(\mathcal{O}_S) =0$.
In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur.
Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$ Catanese, Fabrizio Frapporti, Davide Gleissner, Christian Liu, Wenfei Schütt, Matthias Algebraic Geometry Complex Variables 14J50, 14J80, 14J27, 14H30, 14F99, 32L05, 32M99, 32Q15, 32Q55 In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $κ(S)=1$), in the initial case $ χ(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components. |
| title | On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$ |
| topic | Algebraic Geometry Complex Variables 14J50, 14J80, 14J27, 14H30, 14F99, 32L05, 32M99, 32Q15, 32Q55 |
| url | https://arxiv.org/abs/2408.16936 |