A note on the Nielsen realization problem for Enriques manifolds

TL;DR

This paper establishes a numerical criterion for the Nielsen realization problem specific to Enriques manifolds, leveraging recent advances in hyper-Kähler geometry and group realization.

Contribution

It introduces a new criterion for realizing groups as automorphisms of Enriques manifolds, expanding understanding of the Nielsen realization problem in this context.

Findings

Derived explicit groups that can or cannot be realized as automorphisms of Enriques manifolds.

Applied the criterion to known examples to determine realizability.

Discussed open questions related to the Nielsen realization problem.

Abstract

We give a numerical criterion for the Nielsen realization problem for Enriques manifolds, based on the recent developments on the Birman-Hilden theory for hyper-K\"ahler manifolds and on Nielsen realization for hyper-K\"ahler manifolds. We apply the criterion to known examples of Enriques manifolds to get explicit groups that can be realized or not realized, and comment on questions related to the Nielsen realization problem.