Global quotients among toric deligne–mumford stacks

This work characterizes global quotient stacks—smooth stacks associated to a finite group acting on 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 G0 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.