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arXiv · 1512.07586

A Theory of Stacky Fans

W. D. Gillam·Sam Molcho
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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.

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W. D. Gillam, Sam Molcho. 2015-12-23. A Theory of Stacky Fans. https://arxiv.org/abs/1512.07586

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