Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case

Abstract

Let $k$ be an algebraic closure of finite fields with odd characteristic $p$ and a smooth projective scheme $\mathbf{X}/W(k)$. Let ${\mathbf{X}}^0$ be its generic fiber and $X$ the closed fiber. For ${\mathbf{X}}^0$ a curve Faltings conjectured that semistable Higgs bundles of slope zero over ${\mathbf{X}}_{{\mathbb{C}}_p}^0$ correspond to genuine representations of the algebraic fundamental group of ${\mathbf{X}}_{{\mathbb{C}}_p}^0$ in his $p$-adic Simpson correspondence. This paper intends to study the conjecture in the characteristic $p$ setting. Among other results, we show that isomorphism classes of rank two semistable Higgs bundles with trivial chern classes over $X$ are associated to isomorphism classes of two dimensional genuine representations of ${\mathbf{X}}^0$ and the image of the association contains all irreducible crystalline representations. We introduce intermediate notions strongly semistable Higgs bundles and quasi-periodic Higgs bundles between semistable Higgs bundles and representations of algebraic fundamental groups. We show that quasi-periodic Higgs bundles give rise to genuine representations and strongly Higgs semistable are equivalent to quasi-periodic. We conjecture that a Higgs semistable bundle is indeed strongly Higgs semistable.

18 pages

References in corpus (2)

• An inverse Cartier transform via exponential in positive characteristic
• Higgs bundles over the good reduction of a quaternionic Shimura curve

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