Global quotients among toric Deligne-Mumford stacks

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.