Logarithmic Cartier Transform

TL;DR

This paper extends the Cartier transform to log smooth schemes in positive characteristic, establishing functors between modules with Higgs fields and integrable connections, and generalizing the classical Cartier descent to the logarithmic setting.

Contribution

It generalizes the Cartier transform to log smooth schemes, introduces indexed structures to refine topoi and crystals, and proves an equivalence of categories using the Azumaya property.

Findings

Constructed crystalline-like topoi and subcategories of crystals.

Established a fully faithful functor between modules with Higgs fields and integrable connections.

Generalized Cartier descent to the logarithmic setting.

Abstract

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 $S=\operatorname{Spec}k$ with the trivial log structure. For a log smooth scheme $X$ over $S,$ we obtain, under the assumption that the exact relative Frobenius lifts to the Witt vectors, 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.