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
Received: 2017-06-08
Revised : 2018-08-29
Accepted: 2018-09-25
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