On the cohomologically trivial automorphisms of elliptic surfaces I: $\chi(S)=0$

TL;DR

This paper investigates the structure of cohomologically trivial automorphisms of properly elliptic surfaces with zero Euler characteristic, providing bounds on their size and explicit examples of certain automorphism groups.

Contribution

It classifies the possible finite automorphism groups and bounds the size of infinite automorphism groups for elliptic surfaces with zero Euler characteristic.

Findings

Finite automorphism groups are at most of order 4.

Possible finite groups are Z/2, Z/3, or (Z/2)^2.

Infinite automorphism groups have at most 2 connected components.

Abstract

In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ 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 $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ 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 $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.