Irreducible Holomorphic Symplectic manifolds with an action of $\mathbb Z^4_3 : \mathcal A_6$

Author:

Paola Comparin, Romain Demelle, Pablo Quezada Mora

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)

journal:

date:

2023-05-07 16:00:00

Abstract

H\"ohn and Mason classified the possible symplectic groups acting on an Irreducible Holomorphic Symplectic (IHS) manifold of K3$^{[2]}$-type, finding that $\mathbb Z_3^4 : \mathcal A_6$ is the symplectic group with the biggest order. In this paper, we study the possible IHS manifolds of K3$^{[2]}$-type with a symplectic action of $\mathbb Z_3^4 : \mathcal A_6$ and also admitting a non-symplectic automorphism. We characterize such IHS manifolds. In particular we prove that there exists a IHS manifold of K3$^{[2]}$-type with finite automorphism group of order 174960, the biggest possible order for the automorphism group of a IHS manifold of K3$^{[2]}$-type, and it is the Fano variety of lines of the Fermat cubic fourfold.

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