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

DOI10.4171/JEMS/897zbMATH Open1444.14048arXiv1311.6424MaRDI QIDQ2327708

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Publication date: 15 October 2019

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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 '{e}tale fundamental group $p{i}_1(X_K)$ 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 $leqp$ initiates a semistable Higgs-de Rham flow and thus those of rank $leqp-1$ with trivial Chern classes induce $k$-representations of $p{i}_1(X_K)$. 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.

Full work available at URL: https://arxiv.org/abs/1311.6424

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