A global Torelli theorem for singular symplectic varieties

TL;DR

This paper extends the global Torelli theorem to singular symplectic varieties that admit resolutions, providing new insights into their deformation theory and implications for their birational geometry.

Contribution

It proves an analog of Verbitsky's global Torelli theorem for certain singular symplectic varieties, extending deformation results from smooth to singular cases.

Findings

Established a Torelli theorem for singular symplectic varieties.

Extended local deformation results to non-projective cases.

Provided applications to the classification of birational contractions.

Abstract

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.