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

Guitang Lan Institut fu ̈r Mathematik, Universita ̈t Mainz, Mainz, 55099, Germany Mao Sheng School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China Kang Zuo Institut fu ̈r Mathematik, Universita ̈t Mainz, Mainz, 55099, Germany

Algebraic Geometry mathscidoc:2203.45005

J. Eur. Math. Soc., 21, 3053-3112, 2019.6
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 flow1 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 π_1(X_K ) of the generic fiber of X, after Fontaine-Laffaille and Faltings, where K is the fraction field of W(k). 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.
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@inproceedings{guitang2019semistable,
  title={Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups},
  author={Guitang Lan, Mao Sheng, and Kang Zuo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316104814499940974},
  booktitle={J. Eur. Math. Soc.},
  volume={21},
  pages={3053-3112},
  year={2019},
}
Guitang Lan, Mao Sheng, and Kang Zuo. Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups. 2019. Vol. 21. In J. Eur. Math. Soc.. pp.3053-3112. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316104814499940974.
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