Periodic Higgs subbundles in positive and mixed characteristic

Let $k$ be an algebraically closed field of odd characteristic $p$ and $X$ a proper smooth scheme over the Witt ring $W(k)$. To an object $(M,Fil^{\cdot},\nabla,\Phi)$ in the Faltings category $\mathcal{MF}^{\nabla}_{[0,n]}(X), n\leq p-2$, one associates an \'{e}tale local system $\V$ over the generic fiber of $X$ and a Higgs bundle $(E,\theta)$ over $X$. Our motivation is to find the analogue of the classical Simpson correspondence for the categories of subobjects of $\V$ and $(E,\theta)$. Our main discovery in this paper is the notion of periodic Higgs subbundles, both in positive characteristic and in mixed characteristic. In char $p$, it relies on the inverse Cartier transform constructed by Ogus and Vologodsky in their work on the char $p$ nonabelian Hodge theory. A lifting of the inverse Cartier transform to mixed characteristic is constructed, which is used for the notion of periodicity in mixed characteristic. We show a one to one correspondence between the set of periodic Higgs subbundles of $(E,\theta)$ and the set of \'{e}tale sub local systems of $\V\otimes_{\Z_{p}}\Z_{p^r}$, where $r$ is a natural number. The notion turns out to be useful in applications. We have proven, among other results, that the reduction $(E,\theta)_0$ of $(E,\theta)$ modulo $p$ is Higgs stable, if and only if, the corresponding representation $\V$ is absolutely irreducible over $k$.