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Hecke transformation for orthogonal bundles over curves

Christian Pauly, Hacen Zelaci

TL;DR

This work introduces an orthogonal analogue of the classical Hecke transformation for vector bundles by using Lagrangian $\mathbb{K}[\epsilon]$-submodules of the doubled fiber $E_{2x}$. It establishes that the transformed bundle $\mathcal{H}(E,L)$ remains orthogonal, details a stratified parameter space $\mathrm{O}\Sigma(E_{2x})$ for possible Hecke data, and develops orthogonal Hecke curves via $\mathrm{OGr}(2,E_x)$. The authors prove a Tyurin-type duality for orthogonal bundles, analyze the transformation in low-rank cases, and relate second Stiefel-Whitney classes under Hecke operations. Overall, the paper extends Hecke theory to orthogonal moduli and opens avenues for a parallel treatment of symplectic and anti-invariant bundles, with explicit rank-specific descriptions and a duality framework that mirrors the classical vector-bundle case.

Abstract

Given an orthogonal bundle $E$ over a smooth projective curve $X$ we define a Hecke transformation in the moduli space of orthogonal bundles by performing an elementary transformation with respect to a Lagrangian submodule $L \subset E_{2x}$ at some point $x \in X$. We show that the analogue of Tyurin's duality theorem holds for orthogonal bundles. Special cases of orthogonal bundles of ranks $2,3,4$ and $6$ are studied in detail.

Hecke transformation for orthogonal bundles over curves

TL;DR

This work introduces an orthogonal analogue of the classical Hecke transformation for vector bundles by using Lagrangian -submodules of the doubled fiber . It establishes that the transformed bundle remains orthogonal, details a stratified parameter space for possible Hecke data, and develops orthogonal Hecke curves via . The authors prove a Tyurin-type duality for orthogonal bundles, analyze the transformation in low-rank cases, and relate second Stiefel-Whitney classes under Hecke operations. Overall, the paper extends Hecke theory to orthogonal moduli and opens avenues for a parallel treatment of symplectic and anti-invariant bundles, with explicit rank-specific descriptions and a duality framework that mirrors the classical vector-bundle case.

Abstract

Given an orthogonal bundle over a smooth projective curve we define a Hecke transformation in the moduli space of orthogonal bundles by performing an elementary transformation with respect to a Lagrangian submodule at some point . We show that the analogue of Tyurin's duality theorem holds for orthogonal bundles. Special cases of orthogonal bundles of ranks and are studied in detail.

Paper Structure

This paper contains 22 sections, 21 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Hecke transformation and modules over the ring of dual numbers
  3. Preliminaries on modules over the ring of dual numbers
  4. Parameterizing $\mathbb{K}[\epsilon]$-submodules
  5. Hecke curves revisited
  6. Tangent space to the stratum $\Sigma_i$
  7. Lagrangian $\mathbb{K}[\epsilon]$-submodules
  8. Preliminaries on Lagrangian submodules
  9. Parameterizing Lagrangian submodules
  10. Tangent space to the largest stratum of $\mathrm{O}\Sigma^{(m)}$
  11. Orthogonal Hecke transformations and orthogonal Hecke curves
  12. Proof of Theorem \ref{['maintheorem']}
  13. Orthogonal Hecke transformations and second Stiefel-Whitney class
  14. Orthogonal Hecke curves
  15. Low rank cases
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $L\subset E_{2x}$ be a Lagrangian submodule and denote by $E'$ the kernel subsheaf of $E$ Then $\mathcal{H}(E, L) := E' \otimes \mathcal{O}(x)$ is an orthogonal bundle.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • ...and 36 more