Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities
Simon Crawford, Susan J. Sierra
TL;DR
The paper quantizes the classical McKay correspondence for type A Kleinian surface singularities by constructing a geometric resolution X_q for the quantum invariant ring B_q = A_q^{C_{n+1}} and proving a derived equivalence with the algebraic noncommutative crepant resolution Λ_q = A_q # C_{n+1}. It introduces a graded ring T_q and shows that qgr T_q provides a geometric resolution whose derived category is equivalent to mod-Λ_q, via a tilting object M. The work also develops a noncommutative toric geometric framework, proves relative Serre finiteness and smoothness, and computes the intersection theory of the exceptional locus, recovering an A_n Dynkin diagram. Overall, the paper extends the McKay correspondence to quantum Kleinian singularities, linking algebraic, geometric, and category-theoretic resolutions through a noncommutative toric approach and tilting theory.
Abstract
Let $B = \Bbbk_q[u,v]^{C_{n+1}}$ be a Type $\mathbb{A}_n$ quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution $Λ$, which is an "algebraic" resolution of $B$. We construct a category $\mathcal{X}$ which serves as a "geometric" resolution of $B$ by adapting techniques from quiver GIT and show that $\mathcal{X}$ and $\text{mod-}Λ$ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of $\mathcal{X}$ corresponds to a Type $\mathbb{A}_n$ Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities.