Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$

Gerald Hoehn, Geoffrey Mason

Published 2014-09-21Version 1

We determine the possible finite groups G of symplectic automorphisms of a hyperk\"ahler manifold which is deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that G 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 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.