## Duke Mathematical Journal

### Poisson deformations of affine symplectic varieties

Yoshinori Namikawa

#### Abstract

We prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. For an affine symplectic variety $X$ with a good $C^{*}$-action (where its natural Poisson structure is positively weighted), the following are equivalent.

(1) $X$ has a crepant projective resolution.

(2) $X$ has a smoothing by a Poisson deformation.

A typical example is (the normalization) of a nilpotent orbit closure in a complex simple Lie algebra. By the theorem, one can see which orbit closure has a smoothing by a Poisson deformation.

#### Article information

Source
Duke Math. J., Volume 156, Number 1 (2011), 51-85.

Dates
First available in Project Euclid: 16 December 2010

https://projecteuclid.org/euclid.dmj/1292509118

Digital Object Identifier
doi:10.1215/00127094-2010-066

Mathematical Reviews number (MathSciNet)
MR2746388

Zentralblatt MATH identifier
1208.14028

Subjects
Primary: 14J 14E 32G
Secondary: 14B 32J

#### Citation

Namikawa, Yoshinori. Poisson deformations of affine symplectic varieties. Duke Math. J. 156 (2011), no. 1, 51--85. doi:10.1215/00127094-2010-066. https://projecteuclid.org/euclid.dmj/1292509118

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