Edidin, Rydh — Canonical Reduction Of Stabilizers For Artin Stacks With Good Moduli Spaces (2021)

We present a complete generalization of Kirwan’s partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if \(\mathcal{ X}\) is an irreducible Artin stack with stable good moduli space \(\mathcal{ X}{} \to X\), then there is a canonical sequence of birational morphisms of Artin stacks \(\mathcal{ X}_n \to \mathcal{ X}_{ n-1}{} \to \ldots \to \mathcal{ X}_0 = \mathcal{ X}\) with the following properties: (1) the maximum dimension of a stabilizer of a point of \(\mathcal{ X}_{ k+1}\) is strictly smaller than the maximum dimension of a stabilizer of \(\mathcal{ X}_k\) and the final stack \(\mathcal{ X}_n\) has constant stabilizer dimension; (2) the morphisms \(\mathcal{ X}_{ k+1}{} \to \mathcal{ X}_k\) induce proper and birational morphisms of good moduli spaces \(X_{ k+1}{} \to X_{ k}\). If in addition the stack \(\mathcal{ X}\) is smooth, then each of the intermediate stacks \(\mathcal{ X}_k\) is smooth and the final stack \(\mathcal{ X}_n\) is a gerbe over a tame stack. In this case the algebraic space \(X_n\) has tame quotient singularities and is a partial desingularization of the good moduli space \(X\). When \(\mathcal{ X}\) is smooth our result can be combined with D. Bergh’s recent destackification theorem for tame stacks to obtain a full desingularization of the algebraic space \(X\).