Automorphisms of an irregular surface with low slope acting trivially in cohomology

Abstract

Let $S$ be a complex minimal nonsingular projective irregular surface of general type with $K_S^2 \leq 4_{\mathcal{X}} (\mathcal{O_S})$ and ${}_\mathcal{X} (\mathcal{O_S}) \gt 12$. Then the group of automorphisms of $S$ acts faithfully on the cohomology ring $H^* (S, \mathbb{Q})$ with the exceptional case that $S$ is as in [Ca3, Theorem 2.5].