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

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 ${\pi }_1(X_K)$ 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 $\le p$ initiates a semistable Higgs-de Rham flow and thus those of rank $\le p-1$ 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.

60 pages (Updated Version). Subsume the manuscript with title "Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case", arXiv:1210.8280. Accepted for publication in "Journal of European Mathematical Society"

References in corpus (7)

• Semistable modules over Lie algebroids in positive characteristic
• Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case
• Semistable Higgs bundles of small ranks are strongly Higgs semistable
• Nonabelian Hodge theory in positive characterstic via exponential twisting
• On a Conjecture of Lan-Sheng-Zuo on Semistable Higgs Bundles: Rank 3 Case
• Uniformization of $p$-adic curves via Higgs-de Rham flows
• Periodic Higgs subbundles in positive and mixed characteristic

Cited by in corpus (4)

• Lifting the Cartier transform of Ogus-Vologodsky modulo $p^n$
• On a Conjecture of Lan-Sheng-Zuo on Semistable Higgs Bundles: Rank 3 Case
• Parallel transport for vector bundles on p-adic varieties
• Uniformization of $p$-adic curves via Higgs-de Rham flows