A finiteness theorem on symplectic singularities. (15th April 2016)

• Record Type: Journal Article
• Authors: Namikawa, Yoshinori
• Is Part Of: Compositio mathematica. Volume 152:Number 6(2016:Nov.)
• Page Start: 1225
• Page End: 1236
• Publication Date: 2016-04-15
• Subjects: 14D15, -- 14J17, -- 14J45, -- 32S30, -- 53D10, -- 53D17 (primary); conical symplectic variety, -- Poisson deformation, -- contact Fano orbifold; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X16007387
• Languages: English
• ISSNs: 0010-437X

Abstract

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.