Singularities of Foliations and Good Moduli Spaces of Algebraic Stacks

TL;DR

This paper extends the theory of singularities from foliations to algebraic stacks, establishing conditions under which good moduli spaces exist and analyzing the structure of stable points.

Contribution

It introduces definitions of canonical and log-canonical singularities for algebraic stacks and links these to the existence of good moduli spaces and the structure of stabiliser groups.

Findings

Log-canonical singularities imply finite extension of algebraic tori in stabilisers.

Existence of good moduli spaces at points with log-canonical singularities.

Stable points form a non-empty locus when singularities are canonical.

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.