## Kyoto Journal of Mathematics

### Poisson deformations of affine symplectic varieties, II

Yoshinori Namikawa

#### Abstract

This article is a continuation of previous work, which has the same title. Let $Y$ be an affine symplectic variety with a $\mathbf{C}^{*}$-action with positive weights, and let $\pi\dvtx X\to Y$ be its crepant resolution. Then $\pi$ induces a natural map $\operatorname {PDef}(X)\to\operatorname{PDef}(Y)$ of Kuranishi spaces for the Poisson deformations of $X$ and $Y$. In Part I, we proved that $\operatorname{PDef}(X)$ and $\operatorname{PDef}(Y)$ are both nonsingular, and this map is a finite surjective map. In this article (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation for this article. As an application, we construct explicitly the universal Poisson deformation of the normalization $\tilde{O}$ of a nilpotent orbit closure $\bar{O}$ in a complex simple Lie algebra when $\tilde{O}$ has a crepant resolution.

#### Article information

Source
Kyoto J. Math., Volume 50, Number 4 (2010), 727-752.

Dates
First available in Project Euclid: 29 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1291041216

Digital Object Identifier
doi:10.1215/0023608X-2010-012

Mathematical Reviews number (MathSciNet)
MR2740692

Zentralblatt MATH identifier
1211.14040

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

#### Citation

Namikawa, Yoshinori. Poisson deformations of affine symplectic varieties, II. Kyoto J. Math. 50 (2010), no. 4, 727--752. doi:10.1215/0023608X-2010-012. https://projecteuclid.org/euclid.kjm/1291041216

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