Non-commutative resolutions of quotient singularities

TL;DR

This paper extends the theory of non-commutative resolutions from finite to reductive groups, showing their existence for quotient singularities and providing new examples, including determinantal varieties.

Contribution

It generalizes non-commutative resolution results to reductive groups and introduces algebraic methods that do not rely on commutative resolutions.

Findings

Reductive group quotient singularities admit non-commutative resolutions.

Existence of twisted non-commutative crepant resolutions for a broad class of singularities.

New cases of non-commutative crepant resolutions for determinantal varieties.

Abstract

In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. Moreover we exhibit a large class of such singularities which have (twisted) non-commutative crepant resolutions. We discuss a number of examples, both new and old, that can be treated using our methods. Notably we prove that twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symmetric matrices. In contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity.