arXiv subjects

Sami Fersi

Publications and source records attributed to Sami Fersi.

3 recordsLinked to original sources

arXiv2026

A Topos-Theoretic Approach to the Logarithmic Cartier Transform

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.

math.AG↗

arXiv2026

Logarithmic Cartier Transform

We generalize the Cartier transform of Ogus and Vologodsky to log smooth schemes. More precisely, we generalize a local version of this transform, due to Shiho, and a topos-theoretic version, due to Oyama. Let $k$ be a perfect field of positive characteristic $p$ and equip $\operatorname{Spec}k$ with the trivial log structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ where $S$ is log flat and locally of finite type over $\operatorname{Spec}k,$ we obtain, under the assumption that the exact relative Frobenius lifts over the Witt vectors of $k,$ a fully faithful functor from the category of quasi-coherent modules on the base change $X'=X\times_{S,F_S}S$ of $X$ by the Frobenius $F_S$ of $S,$ equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on $X$ equipped with a quasi-nilpotent integrable connection. In another direction, we construct crystalline-like topoi and subcategories of crystals $\mathcal{C}'$ and $\underline{\mathcal{C}},$ equivalent respectively to modules with Higgs fields and integrable connections, and a fully faithful functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}}.$ Since the Frobenius morphism is not, in general, flat in the log smooth setting, it is not clear that these functors are essentially surjective. To address this issue, we refine the topoi and crystals mentioned above by endowing them with an indexed structure, inspired by Lorenzon's extension of Cartier descent to smooth logarithmic schemes. Using the Azumaya property of the ring of logarithmic differential operators, we then obtain an equivalence between the corresponding categories of indexed crystals, thereby generalizing the Cartier transform.

math.AG↗

arXiv2026

Local Logarithmic Cartier Transform

This article is the first of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. In this article, we generalize a local version, due to Shiho, of the Cartier transform to log smooth schemes. More precisely, let $k$ be a perfect field of positive characteristic and equip $\operatorname{Spec}k$ with the trivial logarithmic structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ where $S$ is log flat and locally of finite type over $\operatorname{Spec}k,$ we obtain, under the assumption that the exact relative Frobenius lifts over the Witt vectors of $k$ to a morphism between log smooth schemes over $S,$ a fully faithful functor from the category of quasi-coherent modules on the base change $X'=X\times_{S,F_S}S$ of $X$ by the Frobenius $F_S$ of $S,$ equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on $X$ equipped with a quasi-nilpotent integrable connection. For this, we prove a log flat descent theorem for morphisms, based on previous work by Kato.

math.AG↗