On logarithmic nonabelian Hodge theory of higher level in characteristic $p$

Abstract

Given a natural number $m$ and a log smooth integral morphism $X\to S$ of fine log schemes of characteristic $p>0$ with a lifting of its Frobenius pull-back $X^{\prime }\to S$ modulo $p^2$, we use indexed algebras ${\mathcal{A}}_X^{gp}$, ${\mathcal{B}}_{X/S}^{(m+1)}$ of Lorenzon-Montagnon and the sheaf ${\mathcal{D}}_{X/S}^{(m)}$ of log differential operators of level $m$ of Berthelot-Montagnon to construct an equivalence between the category of certain indexed ${\mathcal{A}}_X^{gp}$-modules with ${\mathcal{D}}_{X/S}^{(m)}$-action and the category of certain indexed ${\mathcal{B}}_{X/S}^{(m+1)}$-modules with Higgs field. Our result is regarded as a level $m$ version of some results of Ogus-Vologodsky and Schepler.