Comparing the Kirwan and noncommutative resolutions of quotient varieties
Špela Špenko, Michel Van den Bergh
TL;DR
The paper investigates how noncommutative crepant resolutions (NCCRs) of quotient varieties $X/\\!/G$ relate to Kirwan resolutions. It proves that $D(\\End_X({\mathcal U})^G)$ embeds into the derived category of the Kirwan resolution $D({\bf X}/G)$ and, under CM and generator in codimension-one conditions, promotes this embedding to a full semi-orthogonal decomposition with additional components given by endomorphism algebras on stabilized centers. For abelian $G$ explicit SODs decompose $D({\bf X}/G)$ into $D(\\Lambda)$ plus multiple copies of $D( Z_{ji}/(G/H_{ji}) )$, while in general these extra pieces are described by Azumaya algebras over smooth DM-stacks, revealing a gerby structure. The work combines Reichstein and Kirwan transforms with Luna slice methods to provide a robust, local-to-global framework for understanding NCCRs within the Kirwan resolution, including concrete linear-case analyses, generation results, and Morita-theoretic interpretations. Overall, the results bridge NCCRs with geometric resolutions, yielding explicit decompositions and Morita descriptions that illuminate the categorical landscape of quotient singularities and their resolutions.
Abstract
Let a reductive group $G$ act on a smooth variety $X$ such that a good quotient $X{/\!\!/}G$ exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of $X{/\!\!/} G$, obtained from a $G$-equivariant vector bundle on $X$, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of $X{/\!\!/} G$. In fact the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne-Mumford stacks.