ABSTRACT. We construct a new twenty-dimensional family of projective eight-dimen-

We systematically study the moduli theory of singular symplectic varieties which have a resolution by an irreducible symplectic manifold and prove an analog of Verbitsky's global Torelli theorem. In place of twistor lines, Verbitsky's work on ergodic complex structures provides the essential global input. On the one hand, our deformation theoretic results are a further generalization of Huybrechts' theorem on deformation equivalence of birational hyperkähler manifolds to the context of singular symplectic varieties. On the other hand, our global moduli theory provides a framework for understanding and classifying the symplectic singularities that arise from birational contractions of irreducible symplectic manifolds, and there are a number of applications to K3 [n] -type varieties.

No abstract

If (G, V ) is a polar representation with Cartan subspace c and Weyl group W , it is shown that there is a natural morphism of Poisson schemes c ⊕ c * /W → V ⊕ V * / / /G. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if (G, V ) is visible. The conjecture is proved for visible stable locally free polar representations and some other examples.

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of K3 [n] -type varieties.