Tamely ramified geometric Langlands correspondence in positive characteristic

Shen, Shiyu

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https://hdl.handle.net/2142/105797 Copy

Description

Title

Tamely ramified geometric Langlands correspondence in positive characteristic

Author(s)

Shen, Shiyu

Issue Date

2019-07-09

Director of Research (if dissertation) or Advisor (if thesis)

Nevins, Thomas

Doctoral Committee Chair(s)

Haboush, William

Committee Member(s)

• McGerty, Kevin
• Dodd, Christopher

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Hitchin systems, geometric Langlands correspondence

Abstract

We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k), where k is 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 GLn(k) at each marked point. We denote by Bun(n,P) the moduli stack of (quasi-)parabolic vector bundles on X, and by Loc(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(QCoh(Loc^0(n,P))) of quasi-coherent sheaves on an open substack Loc^0(n, P) of Loc(n,P), and the bounded derived category D^b(D^0Bun(n,P)-mod) of D^0Bun(n,P)-modules, where D^0Bun(n,P) is a localization of DBun(n,P) the sheaf of crystalline differential operators on Bun(n,P). Thus we extend the work of Bezrukavnikov-Braverman [8] 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.

Graduation Semester

2019-08

Type of Resource

text

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http://hdl.handle.net/2142/105797

Copyright and License Information

2019 by Shiyu Shen. All rights reserved.

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