University of Illinois Urbana-Champaign

Tamely ramified geometric Langlands correspondence in positive characteristic

Shen, Shiyu

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SHEN-DISSERTATION-2019.pdf

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

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
Date of Ingest
2019-11-26T20:49:23Z
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
Permalink
http://hdl.handle.net/2142/105797
Copyright and License Information
2019 by Shiyu Shen. All rights reserved.

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