Comparing the Kirwan and noncommutative resolutions of quotient varieties

TL;DR

This paper compares noncommutative crepant resolutions and Kirwan resolutions of quotient varieties, showing how their derived categories relate through embeddings and semi-orthogonal decompositions.

Contribution

It demonstrates that the derived category of an NCCR can be embedded into the Kirwan resolution's derived category, with a semi-orthogonal decomposition involving Azumaya algebras.

Findings

Derived category of NCCR embeds into Kirwan resolution

Semi-orthogonal decomposition includes Azumaya algebra categories

Establishes a categorical relationship between two types of 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.