Duality for dormant opers

TL;DR

This paper establishes a duality between dormant $ ext{sl}_n$-opers and $ ext{sl}_{p-n}$-opers on stable curves in characteristic p, and proves the uniqueness of dormant $ ext{sl}_{p-1}$-opers.

Contribution

It introduces a duality theory for dormant opers of different ranks and proves the uniqueness of a specific dormant oper in characteristic p.

Findings

Duality between dormant $ ext{sl}_n$-opers and $ ext{sl}_{p-n}$-opers.

Existence of a unique dormant $ ext{sl}_{p-1}$-oper.

Applicable to fixed pointed stable curves in characteristic p.

Abstract

In the present paper, we prove that on a fixed pointed stable curve of characteristic $p>0$, there exists a duality between dormant $\mathfrak{sl}_n$-opers ($1 < n <p-1$) and dormant $\mathfrak{sl}_{(p-n)}$-opers. Also, we prove that there exists a unique (up to isomorphism) dormant $\mathfrak{sl}_{(p-1)}$-oper on a fixed pointed stable curve of characteristic $p>0$.