Inhomogeneous Kleinian singularities and quivers
Volume 113 / 2017
Abstract
The purpose of this article is to generalize a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of types $A_r$, $D_r$, $E_6$, $E_7$ and $E_8$ to the singularities of inhomogeneous types $B_r$, $C_r$, $F_4$ and $G_2$ defined in 1978 by P. Slodowy. Let $\Gamma$ be a finite subgroup of $\mathrm{SU}_2$. Then $\mathbb{C}^2/\Gamma$ is a simple singularity of type $\Delta(\Gamma)$. By studying the representation space of a quiver defined from $\Gamma$ via the McKay correspondence, and a well chosen finite subgroup $\Gamma’$ of $\mathrm{SU}_2$ containing $\Gamma$ as normal subgroup, we will use the symmetry group $\Omega=\Gamma’/\Gamma$ of the Dynkin diagram $\Delta(\Gamma)$ and explicitly compute the semiuniversal deformation of the singularity $(\mathbb{C}^2/\Gamma,\Omega)$ of inhomogeneous type. The fibers of this deformation are all equipped with an induced $\Omega$-action. By quotienting we obtain a deformation of a singularity $\mathbb{C}^2/\Gamma’$ with some unexpected fibers.
