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A finiteness theorem on symplectic singularities

Part of: Symplectic geometry, contact geometry Singularities Surfaces and higher-dimensional varieties Families, fibrations

Published online by Cambridge University Press:  15 April 2016

Yoshinori Namikawa
Yoshinori Namikawa*
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan email namikawa@math.kyoto-u.ac.jp
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Abstract

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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.

Keywords

conical symplectic varietyPoisson deformationcontact Fano orbifold

MSC classification

Primary: 14D15: Formal methods; deformations 14J17: Singularities 14J45: Fano varieties 32S30: Deformations of singularities; vanishing cycles 53D10: Contact manifolds, general 53D17: Poisson manifolds; Poisson groupoids and algebroids
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 6 , June 2016 , pp. 1225 - 1236
Copyright
© The Author 2016 

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