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A proof of Dolbeault geometric Langlands for $\GL_2$ with reduced spectral curves

Yukinobu Toda

TL;DR

The paper proves the Dolbeault geometric Langlands correspondence for GL_2 over the locus of reduced spectral curves by employing limit categories that provide a robust classical limit beyond the elliptic locus. It builds a Fourier–Mukai transform via the Arinkin sheaf, proves Wilson/Hecke compatibility of the transform, and identifies a Whittaker normalization connecting the vacuum Poincaré kernel to a left adjoint s_{!} of the Hitchin section, enabling a complete GL_2 argument on non-quasi-compact stacks. The work also outlines a strategy for extending to GL_r and discusses obstructions, while leveraging results on compactified Jacobians of reduced curves (MRVF2) to secure the critical normalization step. Overall, the paper demonstrates that limit-categorical methods can realize a Dolbeault Langlands equivalence beyond the elliptic locus and provides a concrete path toward generalization in non-quasi-compact settings. The results illuminate deep links between categorical Donaldson–Thomas theory and geometric Langlands, offering a framework for further extensions via parabolic induction and stability-parameter variations.

Abstract

In our previous paper with Tudor Pădurariu, we introduced the notion of limit categories for moduli stacks of Higgs bundles and formulated the Dolbeault geometric Langlands correspondence. These limit categories are expected to provide an effective ``classical limit'' of the categories of D-modules on the moduli stack of bundles, and our formulation links categorical Donaldson-Thomas theory with the geometric Langlands correspondence. In this paper, we prove the above Dolbeault geometric Langlands correspondence for $\GL_2$ over the locus in the Hitchin base where the spectral curves are reduced. This is the first non-trivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential to the formulation and proof of the correspondence. Our approach also outlines a strategy for proving the correspondence in greater generality and explains the current obstructions to such an extension.

A proof of Dolbeault geometric Langlands for $\GL_2$ with reduced spectral curves

TL;DR

The paper proves the Dolbeault geometric Langlands correspondence for GL_2 over the locus of reduced spectral curves by employing limit categories that provide a robust classical limit beyond the elliptic locus. It builds a Fourier–Mukai transform via the Arinkin sheaf, proves Wilson/Hecke compatibility of the transform, and identifies a Whittaker normalization connecting the vacuum Poincaré kernel to a left adjoint s_{!} of the Hitchin section, enabling a complete GL_2 argument on non-quasi-compact stacks. The work also outlines a strategy for extending to GL_r and discusses obstructions, while leveraging results on compactified Jacobians of reduced curves (MRVF2) to secure the critical normalization step. Overall, the paper demonstrates that limit-categorical methods can realize a Dolbeault Langlands equivalence beyond the elliptic locus and provides a concrete path toward generalization in non-quasi-compact settings. The results illuminate deep links between categorical Donaldson–Thomas theory and geometric Langlands, offering a framework for further extensions via parabolic induction and stability-parameter variations.

Abstract

In our previous paper with Tudor Pădurariu, we introduced the notion of limit categories for moduli stacks of Higgs bundles and formulated the Dolbeault geometric Langlands correspondence. These limit categories are expected to provide an effective ``classical limit'' of the categories of D-modules on the moduli stack of bundles, and our formulation links categorical Donaldson-Thomas theory with the geometric Langlands correspondence. In this paper, we prove the above Dolbeault geometric Langlands correspondence for over the locus in the Hitchin base where the spectral curves are reduced. This is the first non-trivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential to the formulation and proof of the correspondence. Our approach also outlines a strategy for proving the correspondence in greater generality and explains the current obstructions to such an extension.
Paper Structure (48 sections, 54 theorems, 579 equations)

This paper contains 48 sections, 54 theorems, 579 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Summary of the ingredients of Theorem \ref{['thm:intro']}
  4. History
  5. Outline of the proof of Theorem \ref{['thm:intro']}
  6. Toward a proof of Conjecture \ref{['conj:intro']}
  7. Acknowledgements
  8. Notation and Conventions
  9. Notations for (ind)coherent sheaves
  10. Notation on Gm-weights
  11. Preliminary
  12. Moduli stacks of Higgs bundles
  13. Moduli stacks of semistable Higgs bundles
  14. Open locus of Hitchin base
  15. Limit categories
  16. ...and 33 more sections

Key Result

Theorem 1.2

(Theorem thm:gl2) For $G=\operatorname{GL}_2$, Conjecture conj:intro holds over the locus $\mathrm{B}^{\mathrm{red}} \subset \mathrm{B}$.

Theorems & Definitions (124)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 114 more