Very symmetric hyper-K\"ahler fourfolds

TL;DR

This paper classifies finite groups acting symplectically on hyper-K"ahler fourfolds of K3$^{[2]}$-type, extending known maximal group actions and providing examples of such fourfolds with specific symmetries.

Contribution

It introduces a lattice-theoretic approach to classify all finite groups containing maximal symplectic groups acting on K3$^{[2]}$-type fourfolds and constructs explicit examples.

Findings

Classification of groups containing maximal symplectic groups

Identification of fourfolds with specific group actions

Use of lattice theory for classification

Abstract

G. H\"ohn and G. Mason classified all finite groups acting faithfully and symplectically on a hyper-K{\"a}hler fourfolds of type K3$^{[2]}$. There are 15 maximal among them, call them $\widetilde{G}_1,\ldots, \widetilde{G}_{15}$. Every manifold of type K3$^{[2]}$ admitting an action of $\widetilde{G}_i$ for some $i$ must necessarily have Picard rank 21 which is maximal. This fact allows us to use lattice-theoretic methods to classify all the finite groups $G$ acting faithfully on a hyper-K{\"a}hler fourfold of type K3$^{[2]}$ $X$ such that $G$ contains $\widetilde{G}_i$ as a proper subgroup and $\widetilde{G}_i$ acts symplectically on $X$. We also describe examples of fourfolds of K3$^{[2]}$-type admitting an action of such groups.