Duke Mathematical Journal

Poisson deformations of affine symplectic varieties

Yoshinori Namikawa

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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

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

First available in Project Euclid: 16 December 2010

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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