Singularity categories of deformations of 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.