Singularity categories of deformations of Kleinian singularities

TL;DR

This paper classifies the singularity categories of deformations of Kleinian singularities, linking them to Dynkin subgraphs, and extends intersection theory from geometric McKay correspondence to a noncommutative framework.

Contribution

It determines the singularity categories of deformed Kleinian singularities and relates them to Dynkin subgraphs, generalizing previous results and extending intersection theory.

Findings

Singularity categories correspond to Dynkin subgraphs.

Generalization of intersection theory in noncommutative setting.

Extension of known results on Kleinian singularities.

Abstract

Let $G$ be a finite subgroup of $\text{SL}(2,\Bbbk)$ and let $R = \Bbbk[x,y]^G$ be the coordinate ring of the corresponding Kleinian singularity. In 1998, Crawley-Boevey and Holland defined deformations $\mathcal{O}^\lambda$ of $R$ parametrised by weights $\lambda$. In this paper, we determine the singularity categories $\mathcal{D}_{\text{sg}}(\mathcal{O}^\lambda)$ of these deformations, and show that they correspond to subgraphs of the Dynkin graph associated to $R$. This generalises known results on the structure of $\mathcal{D}_{\text{sg}}(R)$. We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting.