Birational geometry of singular symplectic varieties and a global Torelli theorem

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 example
of Verbitsky's technique which might carry the name "how to deduce beautiful consequences from ugly
behavior of moduli spaces''.