Moduli spaces of cubic threefolds and of irreducible holomorphic symplectic manifolds

In this talk, based on joint work of the author and S. Boissière and A. Sarti, I will first describe an isomorphism between the moduli space of smooth cubic threefolds, as described by Allcock, Carlson and Toledo, and the moduli space of fourfolds of K3^[2] -type with a special non-symplectic automorphism of order three; then, I will show some consequences of this isomorphism concerning degenerations of non-symplectic automorphisms corresponding to degenerations to singular cubic threefolds. Finally we will explore possible generalizations of this problem to higher dimensional irreducible holomorphic symplectic manifolds and to other moduli spaces related to cubic threefolds.