Hyperkähler fourfolds with an action of A7

Compact Hyperkähler manifolds (also called Irreducible Holmorphic Symplectic) are of great interest because of the famous classification theorem of complex manifold with vanishing first Chern class and for the very rich structure on the second cohomology. Most of the geometrical information is in fact recovered by the data of the Hodge decomposition in degree two and its relation with the lattice structure on the integral second cohomology. Due to their rigidity, very few examples have been found and it is very not known if indeed there are others. A family of Hyperkähler fourfolds arise as double covers of EPW sextics associated with a certain Lagrangian space. In a joint work with T. Wawak we construct two quite explicit examples of such fourfolds with a symplectic action of the alternating group A7. These are polarized manifolds with polarization 2 and from a classification of the possible automorphisms only two such examples are expected, but no information on the Picard lattice is available so we are not able to distinguish them yet.