Automorphisms of a surface of general type acting trivially in cohomology
type: Item
instance of: scholarly article
title: Automorphisms of a surface of general type acting trivially in cohomology
zbMATH Open document ID: 1076.14052
author: Jin-Xing Cai
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-03-21T00:00:00Z
review text: This paper studies automorphisms of minimal complex surfaces \(S\) of general type, which act trivially on \(H^2(S, {\mathbb Q})\). The main result states that if \(G\subset \text{ Aut}\,S\) acts trivially on \(H^2(S, {\mathbb Q})\) and \(\chi({\mathcal O}_S)>188\) then \(G\) is isomorphic to a cyclic group of order \(n\) or \(G\) is isomorphic to \( {\mathbb Z_2}\times {\mathbb Z_2}\). The principal ingredients of the proof are \textit{A. Beauville}'s results on the canonical map of surfaces of general type [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)]. The result is proved by first showing that the order of \(G\) as above must be \(\leq 4\) , when the canonical map of \(S\) is generically finite and \(\chi({\mathcal O}_S)\geq 31\). So one is reduced to studying the problem for surfaces which canonical map is composed with a pencil. By Beauville's results for \(\chi({\mathcal O}_S)\) big enough such pencil has general fibre \(f\) with genus \(g\) \(\leq 5\). Then (and this is the bulk of the paper) a case-by-case study for \(g\) yields the result. The case \(g=3\) is the harder to study.
reviewed by: Margarida Mendes Lopes
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14J29
zbMATH DE Number: 2148490
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246671
OpenAlex ID: W2141727303
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DOI: 10.2748/TMJ/1113246671
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description: scientific article; zbMATH DE number 2148490