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On the noncommutative Bondal-Orlov conjecture for some toric varieties (1804.02881v1)

Published 9 Apr 2018 in math.AG and math.RA

Abstract: We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of "weakly symmetric" unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of specific toric NCCRs constructed earlier by the authors.

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