Semistable Higgs bundles and representations of fundamental groups over positive characteristics

Abstract:

My lecture is based on a recent joint paper with G-T. Lan and M. Sheng.

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 $\mathbf{X}$ the closed fiber. For $\mathbf{X}^0$ a curve Faltings conjectured that semistable Higgs bundles of slope zero over $\mathbf{X}^0_{\mathbb{C}_p}$ correspond to genuine representations of the algebraic fundamental group of $\mathbf{X}^0_{\mathbb{C}_p}$ 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 $\mathbf{X}$ are associated to isomorphism classes of two dimensional genuine representations of $\pi_1(\mathbf{X}^0)$ 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.