Deformations of type D Kleinian singularities

Abstract

For $n\geq 4$ 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 $\mathcal{O}^{\boldsymbol{\lambda}}$ constructed by Crawley-Boevey and Holland. For each $\mathcal{O}^{\boldsymbol{\lambda}}$ which is not commutative we shall exhibit an explicit isomorphism $D(q)\cong \mathcal{O}^{\boldsymbol{\lambda}}$ 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 $\mathcal{O}^{\boldsymbol{\lambda}}$ and determine when two $\mathcal{O}^{\boldsymbol{\lambda}}$ are isomorphic.