# The global moduli theory of symplectic varieties

## TL;DR

This paper develops a comprehensive moduli theory for symplectic varieties, extending classical results like the Torelli theorem to a broader, possibly singular, setting, and providing new proofs without relying on hyperkähler metrics.

## Contribution

It introduces the global moduli framework for symplectic varieties and proves a Torelli theorem analogous to the smooth case, broadening the understanding of their geometric structure.

## Key findings

- Established a global Torelli theorem for symplectic varieties.
- Provided a new proof of Verbitsky's Torelli theorem without hyperkähler metrics.
- Extended classical results to singular symplectic varieties.

## Abstract

We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperk\"ahler metric or twistor deformations.