On log local Cartier transform of higher level in characteristic $p$

TL;DR

This paper extends the log local Cartier transform to higher levels in characteristic p, constructing new splitting modules under stronger liftability assumptions and establishing equivalences between D-modules and Higgs modules in the log scheme setting.

Contribution

It introduces a new splitting module over a scalar extension under stronger liftability conditions, generalizing previous results to log schemes and higher levels.

Findings

Constructed a new splitting module over a scalar extension.

Established an equivalence between certain D-modules and Higgs modules.

Discussed compatibility with log Frobenius descent and relations to previous constructions.

Abstract

In our previous paper, given an integral log smooth morphism $X\to S$ of fine log schemes of characteristic $p>0$, we studied the Azumaya nature of the sheaf of log differential operators of higher level and constructed a splitting module of it under an existence of a certain lifting modulo $p^{2}$. In this paper, under a certain liftability assumption which is stronger than our previous paper, we construct another splitting module of our Azumaya algebra over a scalar extension, which is smaller than our previous paper. As an application, we construct an equivalence, which we call the log local Cartier transform of higher level, between certain $\cal D$-modules and certain Higgs modules. We also discuss about the compatibility of the log Frobenius descent and the log local Cartier transform and the relation between the splitting module constructed in this paper and that constructed in the previous paper. Our result can be considered as a generalization of the result of Ogus-Vologodsky, Gros-Le Stum-Quir\'os to the case of log schemes and that of Schepler to the case of higher level.