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A finiteness theorem on symplectic singularities
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 6 / June 2016
- Published online by Cambridge University Press:
- 15 April 2016, pp. 1225-1236
- Print publication:
- June 2016
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- Article
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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.
