Singularities of Foliations and Good Moduli Spaces of Algebraic Stacks

Abstract

Drawing on the theory of Minimal Model Program singularities for foliations, we define relative canonical and log-canonical singularities for algebraic stacks with finite generic stabilisers. We show that if a point has log-canonical singularities, its stabiliser group is a finite extension of an algebraic torus, thus, \'etale locally, the good moduli space exists. If the singularity is canonical, we further show that the locus of stable points is non-empty.