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The global nilpotent cone for universal curves

David Nadler, Zhiwei Yun

Abstract

We construct a conic Lagrangian in the cotangent bundle of the moduli stack of $G$-bundles over the universal curve, restricting to the global nilpotent cone for each curve. It gives rise to a singular support condition suitable for the Betti geometric Langlands correspondence for families of curves and the automorphic gluing functor studied in arXiv: 2105.12318. We also prove a family version of ``local constancy of Hecke operators," generalizing our earlier result.

The global nilpotent cone for universal curves

Abstract

We construct a conic Lagrangian in the cotangent bundle of the moduli stack of -bundles over the universal curve, restricting to the global nilpotent cone for each curve. It gives rise to a singular support condition suitable for the Betti geometric Langlands correspondence for families of curves and the automorphic gluing functor studied in arXiv: 2105.12318. We also prove a family version of ``local constancy of Hecke operators," generalizing our earlier result.
Paper Structure (34 sections, 24 theorems, 74 equations)

This paper contains 34 sections, 24 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Betti geometric Langlands
  3. The universal nilpotent cone
  4. Local constancy of automorphic category
  5. Hecke stability of nilpotent sheaves
  6. Acknowledgments
  7. Generalities on conic Lagrangians
  8. Conic Lagrangians for schemes
  9. Lifting relative Lagrangians
  10. Conic Lagrangians for stacks
  11. Transport of Lagrangians
  12. Relative transport
  13. Global nilpotent cone: recollections
  14. Groups
  15. Marked smooth DM curves
  16. ...and 19 more sections

Key Result

Theorem 1.2.1

There exists a unique closed conic Lagrangian $\mathcal{N}_{\pi}\subset T^*\textup{Bun}_G(\pi)$ such that for any closed point $s\in S$, $p_{s}$ restricted to the fiber $(\mathcal{N}_{\pi})_{s}$ over $s$ gives a set-theoretic bijection $p_{s}: (\mathcal{N}_{\pi})_{s}\to \mathcal{N}_{\mathfrak{X}_{s}

Theorems & Definitions (54)

  • Theorem 1.2.1
  • Conjecture 1.3.1
  • Theorem 1.4.1
  • Definition 2.1.1
  • Lemma 2.1.2
  • proof
  • Lemma 2.2.1
  • proof
  • Proposition 2.2.2
  • proof
  • ...and 44 more