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October, 2019 Numerically trivial automorphisms of Enriques surfaces in characteristic 2
Igor DOLGACHEV, Gebhard MARTIN
J. Math. Soc. Japan 71(4): 1181-1200 (October, 2019). DOI: 10.2969/jmsj/78867886

Abstract

An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second cohomology group (this group modulo torsion subgroup). Extending the results of Mukai and Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\operatorname{Aut}_{\operatorname{ct}}(S)$ of an Enriques surface $S$ is of order $\le 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\mathrm{Aut}_{\operatorname{ct}}(S)$ is a cyclic group of odd order $n \in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order 8 and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also prove that the group $\mathrm{Aut}_{\operatorname{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\le 4$ unless $p = 2$, where $\mathrm{Aut}_{\operatorname{nt}}(S)$ is a subgroup of a 2-elementary group of rank $\le 2$.

Funding Statement

The second author was supported by the DFG Sachbeihilfe LI 1906/3 - 1 “Automorphismen von Enriques Flächen”.

Citation

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Igor DOLGACHEV. Gebhard MARTIN. "Numerically trivial automorphisms of Enriques surfaces in characteristic 2." J. Math. Soc. Japan 71 (4) 1181 - 1200, October, 2019. https://doi.org/10.2969/jmsj/78867886

Information

Received: 21 September 2017; Revised: 12 June 2018; Published: October, 2019
First available in Project Euclid: 20 March 2019

zbMATH: 07174402
MathSciNet: MR4023303
Digital Object Identifier: 10.2969/jmsj/78867886

Subjects:
Primary: 14J28
Secondary: 14J50

Keywords: action of automorphisms on cohomology , Enriques surfaces

Rights: Copyright © 2019 Mathematical Society of Japan

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Vol.71 • No. 4 • October, 2019
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