Canonical reduction of stabilizers for Artin stacks with good moduli spaces

Abstract

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 projective 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 $X$.