# Tamely ramified geometric Langlands correspondence in positive
characteristic

Research paper by **Shiyu Shen**

Indexed on: **29 Oct '18**Published on: **29 Oct '18**Published in: **arXiv - Mathematics - Algebraic Geometry**

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#### Abstract

We prove a version of the tamely ramified geometric Langlands correspondence
in positive characteristic for $GL_n(k)$. Let $k$ be an algebraically closed
field of characteristic $p> n$. Let $X$ be a smooth projective curve over $k$
with marked points, and fix a parabolic subgroup of $GL_n(k)$ at each marked
point. We denote by $\text{Bun}_{n,P}$ the moduli stack of (quasi-)parabolic
vector bundles on $X$, and by $\mathcal{L}oc_{n,P}$ the moduli stack of
parabolic flat connections such that the residue is nilpotent with respect to
the parabolic reduction at each marked point. We construct an equivalence
between the bounded derived category
$D^{b}(\text{Qcoh}({\mathcal{L}oc_{n,P}^{0}}))$ of quasi-coherent sheaves on an
open substack $\mathcal{L}oc_{n,P}^{0}\subset\mathcal{L}oc_{n,P}$, and the
bounded derived category
$D^{b}(\mathcal{D}^{0}_{{\text{Bun}}_{n,P}}\text{-mod})$ of
$\mathcal{D}^{0}_{{\text{Bun}}_{n,P}}$-modules, where
$\mathcal{D}^0_{\text{Bun}_{n,P}}$ is a localization of
$\mathcal{D}_{\text{Bun}_{n,P}}$ the sheaf of crystalline differential
operators on $\text{Bun}_{n,P}$. Thus we extend the work of
Bezrukavnikov-Braverman to the tamely ramified case. We also prove a
correspondence between flat connections on $X$ with regular singularities and
meromorphic Higgs bundles on the Frobenius twist $X^{(1)}$ of $X$ with first
order poles .