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 , 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.