A Theory of Stacky Fans

TL;DR

This paper develops a comprehensive theory of KM fans, a stacky generalization of fans in toric geometry, establishing realization functors to various geometric categories and proving an equivalence with toric DM stacks.

Contribution

It introduces the category of KM fans and constructs realization functors, proving an equivalence with toric Deligne-Mumford stacks in characteristic zero.

Findings

Equivalence between KM fans and toric DM stacks.

Realization functor to log differentiable spaces yields actual spaces.

Characterization of when a map of KM fans is a torsor.

Abstract

We study the category of KM fans - a "stacky" generalization of the category of fans considered in toric geometry - and its various realization functors to "geometric" categories. The "purest" such realization takes the form of a functor from KM fans to the 2-category of stacks over the category of fine fans, in the "characteristic-zero-\'etale" topology. In the algebraic setting, over a field of characteristic zero, we have a realization functor from KM fans to (log) Deligne-Mumford stacks. We prove that this realization functor gives rise to an equivalence of categories between (lattice) KM fans and an appropriate category of toric DM stacks. Finally, we have a differential realization functor to the category of (positive) log differentiable spaces. Unlike the other realizations, the differential realization of a stacky fan is an "actual" log differentiable space, not a stack. Our main results are generalizations of "classical" toric geometry, as well as a characterization of "when a map of KM fans is a torsor". The latter is used to explain the relationship between our theory and the "stacky fans" of Geraschenko and Satriano.