On Kleinian Singularities and Quivers

Abstract

Starting from McKay’s observation on the description of (an essential part of) the representation theory of binary polyhedral groups Γ in terms of extended Coxeter-Dynkin-Witt diagrams \(\tilde \Delta (\Gamma )\) and working in the differential geometric framework of Hyper-Kahler-quotients P.B. Kronheimer was able to give a new construction of the semiuniversal deformations of the Kleinian singularities X = ℂ2/Γ as well as of their simultaneous resolutions ([24], [25], [26]). As far as the deformations were concerned, he already gave a purely algebraic geometric formulation of his results in terms of representations of certain quivers naturally attached to the diagrams \(\tilde \Delta (\Gamma )\). By making use of the invariant-theoretic notion of “linear modification” (cf. Section 6, below) and applying it to Kronheimer’s quiver construction we show here how to obtain a purely algebraic geometric simultaneous resolution as well (Section 7). On the way, we shall take the opportunity to remind the reader of various facts about Kleinian singularities (Section 1), McKay’s observation (Section 2), Symplectic geometry (Section 3), Kronheimer’s work (Section 4), and Quivers (Section 5).