Comparing the Kirwan and noncommutative resolutions of quotient varieties

2019/12/03 by Špenko, Špela, Bergh, Michel Van den
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1912.01689

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.

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• Mutations of noncommutative crepant resolutions in geometric invariant theory
• Derived categories of resolutions of cyclic quotient singularities
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• Derived equivalences by quantization