Irreducible holomorphic symplectic manifolds with an action of ℤ₃⁴ : 𝒜₆

Abstract

Höhn and Mason classified the groups acting symplectically on an irreducible holomorphic symplectic (IHS) manifold of K3^[2]-type, finding that ℤ₃⁴ : 𝒜₆ is the one with the largest order. In this paper we study IHS manifolds of K3^[2]-type with a symplectic action of ℤ₃⁴ : 𝒜₆ which also admit a non-symplectic automorphism. We characterize such IHS manifolds and prove their existence. We also prove that the order of a finite group acting on an IHS manifold of K3^[2]-type is bounded by 174960, this bound is sharp and there is a unique IHS manifold of K3^[2]-type acted by a group of this order, which is the Fano variety of lines of the Fermat cubic fourfold.