Deformations of type D Kleinian singularities

Abstract

For $n\ge 4$ we shall construct a family $D(q)$ of non-commutative deformations of the coordinate algebra of a Kleinian singularity of type $D_n$ depending on a polynomial $q$ of degree $n$. We shall prove that every deformation of a type $D$ Kleinian singularity which is not commutative is isomorphic to some $D(q)$. We shall then consider in type $D$ the family of deformations ${\mathcal{O}}^{\mathbf{\lambda }}$ constructed by Crawley-Boevey and Holland. For each ${\mathcal{O}}^{\mathbf{\lambda }}$ which is not commutative we shall exhibit an explicit isomorphism $D(q)\cong {\mathcal{O}}^{\mathbf{\lambda }}$ for a suitable choice of $q$. This will enable us to prove that every deformation of a Kleinian singularity of type $D_n$ is isomorphic to some ${\mathcal{O}}^{\mathbf{\lambda }}$ and determine when two ${\mathcal{O}}^{\mathbf{\lambda }}$ are isomorphic.

25 pages. Section 2 rewritten, proof of main theorem shortened slightly

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