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 X is an irreducible Artin stack with stable good moduli space X→πX, then there is a canonical sequence of birational morphisms of Artin stacks Xn→Xn−1→⋯→X0=X with the following properties: (1) the maximum dimension of a stabilizer of a point of Xk+1 is strictly smaller than the maximum dimension of a stabilizer of Xk and the final stack Xn has constant stabilizer dimension; (2) the morphisms Xk+1→Xk induce projective and birational morphisms of good moduli spaces Xk+1→Xk. If in addition the stack X is smooth, then each of the intermediate stacks Xk is smooth and the final stack Xn is a gerbe over a tame stack. In this case the algebraic space Xn has tame quotient singularities and is a partial desingularization of the good moduli space X. When 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.