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 hyperkahler 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.

中文翻译：

---

奇异辛簇的全局 Torelli 定理

我们系统地研究了具有不可约辛流形解析度的奇异辛变体的模理论，并证明了 Verbitsky 的全局 Torelli 定理的类比。Verbitsky 在遍历复杂结构方面的工作代替了扭曲线，提供了必要的全局输入。一方面，我们的变形理论结果是 Huybrechts 关于双有理 hyperkahler 流形变形等价的定理在奇异辛变体的背景下的进一步推广。另一方面，我们的全局模理论为理解和分类由不可约辛流形的双有理收缩引起的辛奇点提供了一个框架，并且对 $K3^{[n]}$ 型变体有许多应用。