Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K{3}^{[2]}$

Abstract

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

67 pages, 14 tables, LaTeX. First Revision contains small changes and additions and minor corrections. 2nd revision contains a short postscript. We also added Table 12 in Magma format as well as some of the Magma and Mathematica procedures used for the computations

References in corpus (11)

• Note on Twisted Elliptic Genus of K3 Surface
• K3 Surfaces, N=4 Dyons, and the Mathieu Group M24
• The 290 fixed-point sublattices of the Leech lattice
• On natural deformations of symplectic automorphisms of manifolds of K3^[n] type
• On symmetries of N=(4,4) sigma models on T^4
• Mori cones of holomorphic symplectic varieties of K3 type
• Mathieu Moonshine and the Geometry of K3 Surfaces
• Second Quantized Mathieu Moonshine
• Automorphisms of hyperkähler manifolds
• Towards a classification of symplectic automorphisms on manifolds of $K{3}^{[n]}$ type
• Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds

Cited by in corpus (6)

• The 290 fixed-point sublattices of the Leech lattice
• The Hodge diamond of O'Grady's 6-dimensional example
• K3 String Theory, Lattices and Moonshine
• Symplectic birational transformations of finite order on O'Grady's sixfolds
• On the Genus of the Moonshine Module
• Automorphisms and Periods of Cubic Fourfolds