Appears in collection : Journées de géométrie arithmétique de l'IHÉS

In this talk, we construct a splitting module of the Azumaya algebra of log differential operators of higher level in characteristic $p$ greater than or equal 0 under the assumption of an existence of certain mod $p^2$ lifting. As an application, we construct a Simpson type correspondence in characteristic $p$ greater than or equal 0. This result can be regarded as a generalization of the result of Ogus-Vologodsky and Gros-Le Stum-Quiros to the case of log schemes and that of Schepler to the case of higher level.