Finite groups of symplectic automorphisms of hyperk\"ahler manifolds of type $K3^{[2]}$

TL;DR

This paper classifies finite groups of symplectic automorphisms on hyperk"ahler manifolds of type $K3^{[2]}$, linking them to Mathieu groups and Leech lattice structures, and explores their deformation classes.

Contribution

It provides a complete classification of possible finite symplectic automorphism groups for these manifolds, connecting group actions to lattice theory and Mathieu Moonshine.

Findings

Finite groups are subgroups of $M_{23}$ or specific groups related to Leech lattice.

Identifies maximal groups with these properties.

Determines deformation classes of such group actions.

Abstract

We determine the possible finite groups $G$ of symplectic automorphisms of hyperk\"ahler manifolds which are deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that $G$ has such an action if, and only if, it is isomorphic to a subgroup of either the Mathieu group $M_{23}$ having at least four orbits in its natural permutation representation on $24$ elements, or one of two groups $3^{1+4}{:}2.2^2$ and $3^4{:}A_6$ associated to $\mathcal{S}$-lattices in the Leech lattice. We describe in detail those $G$ which are maximal with respect to these properties, and (in most cases) we determine all deformation equivalence classes of such group actions. We also compare our results with the predictions of Mathieu Moonshine.