  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 \'{e}tale
fundamental group $\pi_1(X_K)$ 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 $\leq p$ initiates a
semistable Higgs-de Rham flow and thus those of rank $\leq p-1$ with trivial
Chern classes induce $k$-representations of $\pi_1(X_K)$. 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.
