Archives

Finite groups of symplectic automorphisms of hyperkahler manifolds of type K3
by Gerald Höhn   Geoffrey Mason

Vol. 14 No. 2 (2019) P.189~P.264
 DOI: https://doi.org/10.21915/BIMAS.2019204 10.21915/BIMAS.2019204

ABSTRACT

We determine the possible finite groups $G$ of symplectic automorphisms of hyperkahler 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.

KEYWORDS
Hyperkähler manifolds, Hilbert schemes, finite group actions, sporadic groups, moonshine

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 53C26, 55M35, 11F50, 14E07

MILESTONES