Torelli theorems for singular symplectic varieties

Verbitsky's Global Torelli theorem has been one of the most important advances in the theory of holomorphic symplectic manifolds in the last years. In a joint work with Ben Bakker (University of Georgia) we prove a version of the Global Torelli theorem for singular symplectic varieties and discuss applications. Symplectic varieties have interesting geometric as well as arithmetic properties, their birational geometry is particularly rich. We focus on birational contractions of symplectic varieties and generalize a number of known results for moduli spaces of sheaves to general deformations.

Our results are obtained through the interplay of Hodge theory, deformation theory, and a further instance of the concept "how to deduce beautiful consequences from ugly behavior of moduli spaces".