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A note on the bundle underlying Opers

Luca Casarin

TL;DR

This note clarifies a fundamental structural fact about Opers: for a simple Lie algebra $\mathfrak g$ and a smooth curve $C$, the $B$-torsor underlying any $\mathfrak g$-Oper is determined by $C$ and is induced from the canonical ${\rm Aut}_C O$-bundle on $C$. The authors develop the Aut$O$/$\mathrm{Aut}^0_O$ framework, build the jet-space toolkit, and construct the canonical Aut$^0_O$-torsor Aut$_C$, then use Opers’ intrinsic rigidity to prove that any oper is isomorphic to a canonical model $\mathfrak F_0 = {\rm Aut}_C \times_{r_O} B$. The approach fills gaps in the literature by detailing the canonical representative for Opers and demonstrating how the underlying $B$-torsor descends to the curve through cocycle computations in the Aut$^0_3 O$–torsor setting. This work strengthens the conceptual foundation for Opers within the geometric Langlands program and related representation-theoretic contexts. The results have implications for understanding descent, canonical forms, and the interaction between curve geometry and oper structures.

Abstract

In this note we write down a proof of the following well known fact, in order to make the literature more transparent. Let $\mathfrak{g}$ be a simple Lie algebra, then for any smooth curve $C$, the bundle underlying any $\mathfrak{g}$-Oper depends only on the curve and it is induced by the canonical $\text{Aut}\, O$ bundle $\text{Aut}_C$ on $C$.

A note on the bundle underlying Opers

TL;DR

This note clarifies a fundamental structural fact about Opers: for a simple Lie algebra and a smooth curve , the -torsor underlying any -Oper is determined by and is induced from the canonical -bundle on . The authors develop the Aut/ framework, build the jet-space toolkit, and construct the canonical Aut-torsor Aut, then use Opers’ intrinsic rigidity to prove that any oper is isomorphic to a canonical model . The approach fills gaps in the literature by detailing the canonical representative for Opers and demonstrating how the underlying -torsor descends to the curve through cocycle computations in the Aut–torsor setting. This work strengthens the conceptual foundation for Opers within the geometric Langlands program and related representation-theoretic contexts. The results have implications for understanding descent, canonical forms, and the interaction between curve geometry and oper structures.

Abstract

In this note we write down a proof of the following well known fact, in order to make the literature more transparent. Let be a simple Lie algebra, then for any smooth curve , the bundle underlying any -Oper depends only on the curve and it is induced by the canonical bundle on .
Paper Structure (11 sections, 9 theorems, 33 equations)

This paper contains 11 sections, 9 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries and notation
  4. Torsors
  5. Aut O, Jet schemes and all that
  6. The groups Aut O, Aut+ O
  7. Jet schemes and the canonical Aut+ O-torsor
  8. Opers
  9. Connections on smooth curves
  10. Opers
  11. Canonical representative for Opers

Key Result

Proposition 2.1.4

Let $H$ be a smooth algebraic group and let $\mathfrak F \to C$ be an $H$ torsor. Then there exists a Zariski open cover $C = \bigcup_i U_i$ together with étale maps $U'_i \to U_i$ such that $U'_i \times_{U_i} \mathfrak F_{|U_i}$ is a trivial torsor. In particular $\mathfrak F$ is étale-locally triv

Theorems & Definitions (30)

  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Proposition 2.1.4
  • proof
  • Lemma 2.1.5
  • proof
  • Proposition 2.1.6
  • proof
  • Corollary 2.1.7
  • ...and 20 more