A finiteness theorem on symplectic singularities

Abstract

An affine symplectic singularity$X$with a good$\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers$N$and$d$, there are only a finite number of conical symplectic varieties of dimension$2d$with maximal weights$N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.