Abstract
We will present joint work with Olivier Debarre, Daniel Huybrechts and
Claire Voisin on the SYZ hyper-Kähler conjecture for fourfolds under
certain topological assumptions.
As application, this proves a conjecture by O'Grady that a
hyper-Kähler fourfold whose cohomology ring is isomorphic to the one
of the Hilbert square of a K3 surface is a deformation of a Hilbert
square.