# Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

### Guitang Lan

Universität Mainz, Germany### Mao Sheng

University of Science and Technology of China, Hefei, China### Kang Zuo

Universität Mainz, Germany

## Abstract

Let $k$ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of *Higgs–de Rham flow* and prove that the category of periodic Higgs–de Rham flows over $X/W(k)$ is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group $π_{1}(X_{K})$ of the generic fiber of $X$, after Fontaine–Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber $X_{k}$ of $X$ of rank $≤p$ initiates a semistable Higgs–de Rham flow and thus those of rank $≤p−1$ with trivial Chern classes induce $k$-representations of $π_{1}(X_{K})$. A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic $p$, it was constructed by Ogus–Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus–Vologodsky correspondence of Shiho.

## Cite this article

Guitang Lan, Mao Sheng, Kang Zuo, Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups. J. Eur. Math. Soc. 21 (2019), no. 10, pp. 3053–3112

DOI 10.4171/JEMS/897