A global Torelli theorem for singular symplectic varieties

Abstract

We systematically study singular symplectic varieties which have a resolution by an irreducible symplectic manifold. Building on their locally trivial deformation theory we develop a moduli theory and prove an analog of Verbitsky's global Torelli theorem. In this way, we create a framework for understanding and classifying the symplectic singularities that arise from birational contractions of irreducible symplectic manifolds. There are also a number of applications to the study of \(K3^{[n]}\)-type varieties. Our deformation theoretic results are a further instance of a generalization of Huybrechts' theorem on deformation equivalence of birational hyperk\"ahler manifolds to the context of singular symplectic varieties. Apart from this and the moduli theory for smooth symplectic varieties, Verbitsky's work on ergodic complex structures is an essential ingredient in the moduli theoretic results we obtain.