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Perverse sheaves with nilpotent singular support on the stack of coherent sheaves on an elliptic curve

Lucien Hennecart

TL;DR

This work establishes a detailed bridge between the geometry of the elliptic global nilpotent cone and the representation-theoretic theory of perverse sheaves on the stack of coherent sheaves. By exploiting Harder–Narasimhan stratifications and induction diagrams, it classifies irreducible components of the global nilpotent cone and proves a unitriangular characteristic cycle correspondence with simple spherical Eisenstein perverse sheaves, yielding a canonical bijection. It further describes which perverse sheaves on the elliptic stack have nilpotent singular support—proved to be precisely the twisted spherical Eisenstein objects—thereby enhancing the geometric Langlands framework for elliptic curves through explicit combinatorial parametrizations and microlocal data. The results connect the elliptic Hall algebra categorification to concrete geometric objects and provide tools for extending local systems across semistable and non-semistable loci, with potential implications for crystal-like structures and Kostka-type multiplicities in this geometric setting.

Abstract

We define a stratification of the moduli stack of coherent sheaves on an elliptic curve which allows us (1) to give an explicit description of the irreducible components of the global nilpotent cone of elliptic curves, (2) to establish an explicit bijection between the simple objects of the category of perverse sheaves defined by Schiffmann to categorify the elliptic Hall algebra (the so-called spherical Eisenstein sheaves) and the irreducible components of the global nilpotent cone and (3) to give an explicit description and parametrization of the perverse sheaves on the moduli stack of coherent sheaves on an elliptic curve having nilpotent singular support. Along the way, we find a combinatorial parametrization of the irreducible components of the semistable locus of the elliptic global nilpotent cone.

Perverse sheaves with nilpotent singular support on the stack of coherent sheaves on an elliptic curve

TL;DR

This work establishes a detailed bridge between the geometry of the elliptic global nilpotent cone and the representation-theoretic theory of perverse sheaves on the stack of coherent sheaves. By exploiting Harder–Narasimhan stratifications and induction diagrams, it classifies irreducible components of the global nilpotent cone and proves a unitriangular characteristic cycle correspondence with simple spherical Eisenstein perverse sheaves, yielding a canonical bijection. It further describes which perverse sheaves on the elliptic stack have nilpotent singular support—proved to be precisely the twisted spherical Eisenstein objects—thereby enhancing the geometric Langlands framework for elliptic curves through explicit combinatorial parametrizations and microlocal data. The results connect the elliptic Hall algebra categorification to concrete geometric objects and provide tools for extending local systems across semistable and non-semistable loci, with potential implications for crystal-like structures and Kostka-type multiplicities in this geometric setting.

Abstract

We define a stratification of the moduli stack of coherent sheaves on an elliptic curve which allows us (1) to give an explicit description of the irreducible components of the global nilpotent cone of elliptic curves, (2) to establish an explicit bijection between the simple objects of the category of perverse sheaves defined by Schiffmann to categorify the elliptic Hall algebra (the so-called spherical Eisenstein sheaves) and the irreducible components of the global nilpotent cone and (3) to give an explicit description and parametrization of the perverse sheaves on the moduli stack of coherent sheaves on an elliptic curve having nilpotent singular support. Along the way, we find a combinatorial parametrization of the irreducible components of the semistable locus of the elliptic global nilpotent cone.

Paper Structure

This paper contains 51 sections, 30 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. The irreducible components of the elliptic global nilpotent cone
  4. Bijectivity of the characteristic cycle map
  5. Perverse sheaves with nilpotent singular support on the stack of coherent sheaves on an elliptic curve
  6. Strategy of proof
  7. Theorem \ref{['mainirr']}
  8. Theorem \ref{['mainbij']}
  9. Theorem \ref{['microlocchar']}
  10. Contents of the paper
  11. Notations and conventions
  12. The moduli stack of coherent sheaves on elliptic curves
  13. Coherent sheaves on a curve
  14. The category and moduli stack of torsion sheaves
  15. Torsion sheaves on a curve
  16. ...and 36 more sections

Key Result

Theorem 1.1

Let $\alpha\in\mathop{\mathrm{\mathbf{Z}}}\nolimits^+$. The irreducible components of $\mathscr{N}_{\alpha}$ are the \begin{tikzpicture}[baseline=(a.base)] \node[inner xsep=0pt,inner ysep=1.5pt] (a) {$\mathscr{N}_{\bm{\alpha},\bm{\lambda}}$}; \draw[line width= 1.5] (a.north west) -- (a.n

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4: 2017bozec
  • Proposition 3.5
  • ...and 46 more