On the noncommutative Bondal-Orlov conjecture for some toric varieties

TL;DR

This paper demonstrates that toric noncommutative crepant resolutions of certain affine GIT quotients are all derived equivalent, providing evidence for a noncommutative version of the Bondal-Orlov conjecture.

Contribution

It proves that all toric NCCRs of affine GIT quotients are derived equivalent to a fixed GIT quotient stack, extending previous geometric results to a noncommutative setting.

Findings

All toric NCCRs of affine GIT quotients are derived equivalent.

Derived equivalence to a fixed GIT quotient stack is established.

Supports a noncommutative extension of the Bondal-Orlov conjecture.

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.