On the Cohomologically Trivial Automorphisms of Elliptic Surfaces I: $\chi(S) = 0$

Abstract

In this first part we describe the group $\operatorname{Aut}_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S) = 1$), in the initial case $\chi(\mathcal{O}_{S}) = 0$.

In particular, in the case where $\operatorname{Aut}_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $\operatorname{Aut}_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2$, $\mathbb{Z}/3$, $(\mathbb{Z}/2)^{2}$. We also show with easy examples that the groups $\mathbb{Z}/2$, $\mathbb{Z}/3$ do effectively occur.

Respectively, in the case where $\operatorname{Aut}_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.