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arXiv · 1610.08430

Singularity categories of deformations of Kleinian singularities

Simon Crawford
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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 $\mathcal{O}^λ$ of $R$ parametrised by weights $λ$. In this paper, we determine the singularity categories $\mathcal{D}_{\text{sg}}(\mathcal{O}^λ)$ 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.

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Simon Crawford. 2019-10-30. Singularity categories of deformations of Kleinian singularities. https://doi.org/10.2140/ant.2020.14.349

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