The category of toric stacks

Abstract

In this paper, we show that there is an equivalence between the 2-category of smooth Deligne-Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine-Illusie plays a central role in obtaining our results.

Smooth toric Deligne-Mumford stacks

“…Later, Iwanari proposed in [Iwa06] a definition of toric triple as an orbifold with a torus action having a dense orbit isomorphic to the torus 1 and he proved that the 2-category of toric triples is equivalent to the 2-category of "toric stacks" (We refer to [Iwa06] for the definition of "toric stacks"). Nevertheless, it is clear that not all toric Deligne-Mumford stacks are toric triples, since some of them are not orbifolds.…”

The Coherent–Constructible Correspondence for Toric Deligne–Mumford Stacks

“…Remark 5.15. In [26], I. Iwanari established an equivalence between the 2-category of toric stacks and the 1-category of stacky fans. In [26,Definition 2.1], N is a free abelian group, so toric stacks in [26] are toric orbifolds.…”

Toric stacks II: Intrinsic characterization of toric stacks

“…This paper, together with its prequel [GS11a], introduces a theory of toric stacks which encompasses and extends the many pre-existing theories in the literature [Laf02,BCS05,FMN10,Iwa09,Sat12,Tyo12]. Recall from [GS11a, Definition 1.1] that a toric stack is defined to be the stack quotient [X/G] of a normal toric variety X by a subgroup G of the torus of X.…”