Year: 2026
Type: preprint
Abstract: This article is the second of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. We generalize a topos-theoretic version of this transform, due to Oyama. Let $k$ be a perfect field of positive characteristic $p$ and equip $S=\operatorname{Spec}k$ with the trivial log structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ we construct crystalline-like ringed topoi $\mathcal{E}'$ and $\underline{\mathcal{E}}$ and subcategories of crystals of quasi-coherent modules $\mathcal{C}'$ and $\underline{\mathcal{C}},$ equivalent respectively, under some lifting assumption, to modules with Higgs fields and integrable connections, both satisfying certain nilpotence conditions, and a morphism of topoi $\underline{\mathcal{E}} \rightarrow \mathcal{E}'.$ We then prove that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and hence induces a functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}},$ generalizing the Cartier transform. We finally use a log flat descent theorem for morphisms, that we proved in the first article, to prove that this functor is fully faithful.
Source: arXiv (Cornell University)
Author: Fersi, Sami
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214187802