Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball (Q2419962)

The first main result of this paper is the proof of an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of fourfolds deformation-equivalent to the Hilbert square of a $K3$ surface with a special non-symplectic automorphism of order three. The main tool for proof is the use of the period map of irreducible holomorphic symplectic manifolds and Allcock-Carlson-Toledo's period map for cubic threefolds. Then the authors study consequences of the proven isomorphism and provide a geometric description of the degenerations of the non-symplectic automorphism of order three. Also they construct a Pfaffian cyclic cubic fourfold and produce a non-natural automorphism of order three on the Hilbert square of a $K3$ surface.