Periodic Higgs subbundles in positive and mixed characteristic

TL;DR

This paper introduces the concept of periodic Higgs subbundles in positive and mixed characteristic settings, establishing a correspondence with étale sub local systems and exploring their stability and irreducibility properties.

Contribution

It develops the notion of periodic Higgs subbundles in both positive and mixed characteristic, extending the inverse Cartier transform and establishing a correspondence with étale sub local systems.

Findings

One-to-one correspondence between periodic Higgs subbundles and étale sub local systems.

Reduction of Higgs bundle modulo p is stable iff the associated representation is absolutely irreducible.

Constructed a lifting of the inverse Cartier transform to mixed characteristic.

Abstract

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 $k$.