Coherent-Constructible Correspondence for Toric Fibrations

Abstract

Let $\Sigma$ be a fan inside the lattice ${\mathbb{Z}}^n$, and $\mathcal{E}:{\mathbb{Z}}^n\to \mathrm{Pic}S$ be a map of abelian groups. We introduce the notion of a principal toric fibration ${\mathcal{X}}_{\Sigma ,\mathcal{E}}$ over the base scheme $S$, relativizing the usual toric construction for $\Sigma$. We show that the category of ind-coherent sheaves on such a fibration is equivalent to the global section of the Kashiwara-Schapira stack twisted by a certain local system of categories with stalk $\mathrm{Ind}\mathrm{Coh}S$. It is a simultaneous generalization of the work of Harder-Katzarkov [HK19] and of Kuwagaki [Kuw20], and should be seen as a family-version of the coherent-constructible correspondence [FLTZ11].

34 pages, 8 figures