Resolution of singularities via Tannaka duality

TL;DR

This paper introduces a new framework using Tannaka duality to produce both smooth and stacky resolutions of Kleinian singularities, extending previous methods to a broader class of cases.

Contribution

It generalizes Abdelgadir and Segal's approach to all Kleinian singularities, creating Clebsch-Gordan varieties and analyzing their stability and resolutions.

Findings

Successfully generalizes to all Kleinian singularities

Constructs and analyzes Clebsch-Gordan varieties

Provides tools for stability analysis in these varieties

Abstract

Resolving finite quotient singularities is a classical problem in algebraic geometry. Traditional methods of Geometric Invariant Theory (GIT) translate the singularity into a quiver representation space and take the GIT quotient with respect to a generic stability parameter. While this approach easily produces smooth resolutions, it fails to produce any stacky resolutions, as quiver representation spaces lack finite stabilizers. This paper provides an alternative framework which produces both smooth and stacky resolutions. Our framework is based on a trick of Abdelgadir and Segal, which deploys Tannaka duality to describe the points of the classifying stack of a finite group in terms of algebraic data. Abdelgadir and Segal successfully pursue this strategy and obtain smooth and stacky resolutions in the Kleinian $ D_4 $ case. We generalize this strategy to all Kleinian singularities and obtain a series of varieties which we refer to as Clebsch-Gordan varieties. We provide tools to work with these Clebsch-Gordan varieties, analyze their stable loci with respect to different stability parameters, and study the Kleinian $ A_n $ and $ D_n $ cases in detail.