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The spectral gluing theorem revisited

Dario Beraldo

Abstract

We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding.

The spectral gluing theorem revisited

Abstract

We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding.

Paper Structure

This paper contains 22 sections, 22 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. The case of $G= \mathrm{GL}_2$
  3. A baby version
  4. The case of general $G$
  5. Automorphic gluing, a preview
  6. Organization of the paper
  7. Acknowledgements
  8. Global complete intersections
  9. Shift of grading
  10. Singular support for global complete intersections
  11. The singular codifferential
  12. Strong spectral gluing
  13. Some preliminary constructions
  14. A technical note.
  15. Constructing the glued DG category
  16. ...and 7 more sections

Key Result

Theorem 1.1.7

For $G=\mathrm{GL}_2$, the DG category ${\operatorname{IndCoh}}_{\mathcal{N}}({{\operatorname{LS}}_G})$ is naturally equivalent to the limit of the diagram \begin{tikzpicture}[scale=1.5] \node (00) at (0,0) {$\QCoh(\LSG)$}; \node (10) at (3,0) {$\QCoh((\LSG)^\wedge_{\LSB})$.}; %%%%\node (01) at (0,1

Theorems & Definitions (60)

  • Example 1.1.5
  • Theorem 1.1.7: Strong spectral gluing for $G=\mathrm{GL}_2$
  • Remark 1.1.8
  • Remark 1.3.2
  • Theorem 1.3.9: Strong spectral gluing
  • Theorem 1.3.11
  • Remark 1.3.12
  • Example 1.4.6
  • Conjecture 1.4.8
  • Example 2.1.2
  • ...and 50 more