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Title: | On the noncommutative Bondal–Orlov conjecture for some toric varieties | Authors: | SPENKO, Spela VAN DEN BERGH, Michel Bell, Jason P. |
Issue Date: | 2022 | Publisher: | SPRINGER HEIDELBERG | Source: | MATHEMATISCHE ZEITSCHRIFT, 300(1), p. 1055-1068 | 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. 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. |
Notes: | Van den Bergh, M (corresponding author), Vrije Univ Brussel, Dept Wiskunde, Pl Laan 2, B-1050 Ixelles, Belgium.; Van den Bergh, M (corresponding author), Univ Hasselt, Dept WNI, Univ Campus, B-3590 Diepenbeek, Belgium. spela.spenko@vuh.ac.be; michel.vandenbergh@uhasselt.be; jpbell@uwaterloo.ca |
Document URI: | http://hdl.handle.net/1942/36126 | ISSN: | 0025-5874 | e-ISSN: | 1432-1823 | DOI: | 10.1007/s00209-021-02910-8 | ISI #: | WOS:000722086200001 | Rights: | © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2023 |
Appears in Collections: | Research publications |
Files in This Item:
File | Description | Size | Format | |
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s00209-021-02910-8.pdf Restricted Access | Published version | 320.69 kB | Adobe PDF | View/Open Request a copy |
arXiv_1804.02881v1 [math.AG] 9 Apr 2018.pdf | Non Peer-reviewed author version | 192.74 kB | Adobe PDF | View/Open |
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