# Some algebraic and geometric properties of hyper-K\"ahler fourfold symmetries

## TL;DR

This paper classifies order 5 nonsymplectic automorphisms on certain hyper-K"ahler fourfolds, computes Lefschetz numbers for automorphisms of generalized Kummer fourfolds, and describes their fixed loci geometry.

## Contribution

It completes the classification of specific automorphisms on hyper-K"ahler fourfolds and analyzes their geometric and topological properties.

## Key findings

- Classification of order 5 nonsymplectic automorphisms completed.
- Lefschetz numbers computed for automorphisms of generalized Kummer fourfolds.
- Descriptions of fixed loci geometry provided.

## Abstract

We complete the classification of order $5$ nonsymplectic automorphisms on hyper-K\"ahler fourfolds deformation equivalent to the Hilbert square of a K3 surface. We then compute the topological Lefschetz number of natural automorphisms of generalized Kummer fourfolds and we describe the geometry of their fix loci.