Automorphisms of a surface of general type acting trivially in cohomology
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.
scientific article; zbMATH DE number 2148490
scientific article