Automorphisms of an irregular surface of general type acting trivially in cohomology. II
This paper continues the classification of projective, non singular complex surfaces \(S\) having a non-trivial subgroup \(G\) of the group of automorphisms of \(S\) that acts trivially on \(H^{2}(S, \mathbb{Q})\). This classification was started by the author in [Tohoku Math. J., II. Ser. 56, No. 3, 341--355 (2004; Zbl 1076.14052)], and continued in [Trans. Am. Math. Soc. 358, No. 3, 1187--1201 (2006; Zbl 1087.14029); Int. J. Math. 17, No. 2, 183--193 (2006; Zbl 1094.14028); Pac. J. Math. 232, No. 1, 43--59 (2007; Zbl 1152.14041); J. Algebra 322, No. 12, 4228--4246 (2009; Zbl 1200.14068); Adv. Stud. Pure Math. 60, 183--194 (2010; Zbl 1214.14034)]. In the first mentioned paper, it is shown that if \(S\) is minimal of general type, and \(\chi (\mathcal O_S)>188\) then a non-trivial subgroup \(G\) of automorphisms of \(S\) acting trivially on \(H^{2}(S, \mathbb{Q})\), satisfies \(|G|\leq 4\). The present paper classifies pairs \((S,G)\) where \(S\) is a minimal projective, non singular, complex surface \(S\) with \(q=2\) and \(p_g>61\) and \(G\) has above has order \(4\). It is shown that such surfaces are isogenous to a product of curves and a very precise description of the possible (and indeed existing) cases is given.
scientific article; zbMATH DE number 6141567