On derived categories of K3 surfaces, symplectic automorphisms and the Conway group

Abstract

In this note we interpret a recent result of Gaberdiel et al [7] in terms of derived equivalences of K3 surfaces. We prove that there is a natural bijection between subgroups of the Conway group $C\!o_0$ with invariant lattice of rank at least four and groups of symplectic derived equivalences of $\mathrm{D}^{\mathrm{b}}(X)$ of projective K3 surfaces fixing a stability condition.

As an application we prove that every such subgroup $G\subset C\!o_0$ satisfying an additional condition can be realized as a group of symplectic automorphisms of an irreducible symplectic variety deformation equivalent to $\mathrm{Hilb}^n(X)$ of some K3 surface.