Comparing the Kirwan and noncommutative resolutions of quotient varieties

Špenko, Špela; Van den Bergh, Michel

doi:10.1515/crelle-2023-0024

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Comparing the Kirwan and noncommutative resolutions of quotient varieties

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Van den Bergh, Michel
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1515/crelle-2023-0024
(Publisher version)

https://doi.org/10.48550/arXiv.1912.01689
(Preprint)

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Citation

Špenko, Š., & Van den Bergh, M. (2023). Comparing the Kirwan and noncommutative resolutions of quotient varieties. Journal für die reine und angewandte Mathematik, 801, 1-43. doi:10.1515/crelle-2023-0024.

Cite as: https://hdl.handle.net/21.11116/0000-000D-B949-1

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.