Automorphisms of fiber surfaces of genus 2, inducing the identity in cohomology

Abstract

Let S S be a complex non-singular projective surface of general type with a genus 2 2 fibration and χ ( O S ) ≥ 5 \chi (\mathcal O_S)\geq 5 . Let G ⊂ Aut ⁡ S G \subset \operatorname {Aut}S be a non-trivial subgroup of automorphisms of S S , inducing trivial actions on H i ( S , Q ) H^i(S,\mathbb {Q}) for all i i . Then | G | = 2 |G|=2 , K S 2 = 4 χ ( O S ) K_S^2=4\chi (\mathcal O_S) and q ( S ) = 1 q(S)=1 . Examples of such surfaces are given.