Global quotients among toric Deligne-Mumford stacks

TL;DR

This paper characterizes when smooth quotient stacks, especially toric Deligne-Mumford stacks, are global quotients by finite groups, using group action properties and combinatorial data.

Contribution

It provides a new characterization of global quotient stacks among smooth quotient stacks and applies this to classify toric DM stacks as global quotients.

Findings

Characterization of global quotients via group action analysis

Application to toric Deligne-Mumford stacks using combinatorial data

Identification of conditions for stacks to be global quotients

Abstract

This work characterizes global quotient stacks---smooth stacks associated to a finite group acting a manifold---among smooth quotient stacks $[M/G]$, where $M$ is a smooth manifold equipped with a smooth proper action by a Lie group $G$. The characterization is described in terms of the action of the connected component $G_0$ on $M$ and is related to (stacky) fundamental group and covering theory. This characterization is then applied to smooth toric Deligne-Mumford stacks, and global quotients among toric DM stacks are then characterized in terms of their associated combinatorial data of stacky fans.