Comparing the Kirwan and noncommutative resolutions of quotient varieties
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866908837839110144 |
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| author | Špenko, Špela Bergh, Michel Van den |
| author_facet | Špenko, Špela Bergh, Michel Van den |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_01689 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Comparing the Kirwan and noncommutative resolutions of quotient varieties Špenko, Špela Bergh, Michel Van den Algebraic Geometry Rings and Algebras Representation Theory 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. |
| title | Comparing the Kirwan and noncommutative resolutions of quotient varieties |
| topic | Algebraic Geometry Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/1912.01689 |