A note on the Nielsen realization problem for hyper-K\"ahler manifolds

TL;DR

This paper addresses the Nielsen realization problem for hyper-K"ahler manifolds, showing that for certain deformation types, the mapping class group's action on second cohomology admits a section, extending known results from K3 surfaces.

Contribution

It provides a solution to the Nielsen realization problem for hyper-K"ahler manifolds using invariants similar to those for K3 surfaces, identifying cases where a section exists.

Findings

The representation of the mapping class group admits a section on its image for some deformation types.

The approach uses invariants analogous to those for K3 surfaces.

The results extend the understanding of the Nielsen realization problem beyond K3 surfaces.

Abstract

We give an answer to the Nielsen realization problem for hyper-K\"ahler manifolds in terms of the same invariant used for K3 surfaces. We determine that, for some of the known deformation types, the representation of the mapping class group on the second cohomology admits a section on its image.