Let be the algebraic closure of a finite field of odd characteristic and a smooth projective scheme over the Witt ring 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 is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group of the generic fiber of , after Fontaine–Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber of of rank initiates a semistable Higgs–de Rham flow and thus those of rank with trivial Chern classes induce -representations of . A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic , 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.
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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. 3053–3112DOI 10.4171/JEMS/897