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'\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 {\cal 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}^{gp}_{X} -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.