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

### Sachio Ohkawa

University of Tokyo, Japan

## 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.

## Cite this article

Sachio Ohkawa, On logarithmic nonabelian Hodge theory of higher level in characteristic $p$. Rend. Sem. Mat. Univ. Padova 134 (2015), pp. 47–91

DOI 10.4171/RSMUP/134-2