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Tamely ramified geometric Langlands correspondence in positive characteristic

Shiyu Shen

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 .

Tamely ramified geometric Langlands correspondence in positive characteristic

Abstract

We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for . Let be an algebraically closed field of characteristic . Let be a smooth projective curve over with marked points, and fix a parabolic subgroup of at each marked point. We denote by the moduli stack of (quasi-)parabolic vector bundles on , and by 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 of quasi-coherent sheaves on an open substack , and the bounded derived category of -modules, where is a localization of the sheaf of crystalline differential operators on . Thus we extend the work of Bezrukavnikov-Braverman to the tamely ramified case. We also prove a correspondence between flat connections on with regular singularities and meromorphic Higgs bundles on the Frobenius twist of with first order poles .

Paper Structure

This paper contains 30 sections, 47 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Geometric Langlands in positive characteristic
  3. Tamely ramified geometric Langlands correspondence
  4. Summary of the proof
  5. Structure of the article
  6. Notations and definitions
  7. Acknowledgements
  8. Spectral data of parabolic Higgs bundles
  9. Basic constructions
  10. The parabolic Hitchin base $B_P$
  11. Spectral data of parabolic Higgs bundles
  12. Azumaya property of differential operators in positive characteristic
  13. Frobenius twist of a $k$-scheme
  14. Azumaya property of differential operators
  15. Differential operators on smooth stacks
  16. ...and 15 more sections

Key Result

Theorem \oldthetheorem

$\Phi_{\mathcal{P}}$ is an equivalence of derived categories.

Theorems & Definitions (86)

  • Theorem \oldthetheorem
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  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 76 more