The Hodge structure of O'Grady's singular symplectic varieties

Starting from a question of Bakker and C. Lehn about the singularities type of O’Grady’s singular moduli spaces, we investigated their Hodge structure. As consequence of our (partial) computation of it, we deduced that O’Grady’s moduli spaces do not have finite quotient singularities. In the first part of this talk I will introduce O'Grady's moduli spaces, placing them in the setting of singular symplectic varieties. I will state our main result on their Hodge structure, and I will underline the relevant properties of it giving consequences on the singularities type of these spaces. In the second part I will go into the techniques we used for the (partial) computation of the Hodge structure, and I will present a strategy to complete the computation. This is a joint project with Franco Giovenzana