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

2013/11/21 by Lan, Guitang, Sheng, Mao, Zuo, Kang · 1 citation
#14D07 #14F30 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1311.6424

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 π₁(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 π₁(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.

Cited by

• Nonlinear Hodge correspondence in positive characteristic

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