Real line subbundles of real bundles on curves

TL;DR

This paper investigates the action of real structures on maximal line subbundles of stable real rank 2 bundles on genus 2 curves, extending classical results with modern algebraic and geometric techniques.

Contribution

It describes the Galois action on line subbundles of stable real bundles on genus 2 curves, extending classical work with new algebraic and geometric insights.

Findings

Describes the Galois action on line subbundles of rank 2 bundles

Extends classical results of Newstead to real algebraic setting

Provides results on real line subbundles in higher genus

Abstract

For a stable real bundle $E$ of rank $2$ and degree $1$ on a real genus $2$ curve, we describe the action of the real structure of the curve on the set of $4$ maximal line subbundles of degree $0$ of $E$. This describes the Galois action on the set of lines through a real point in the moduli space of such bundles, and is a real algebraic extension of classical work of Newstead. Our proof is an application of techniques of Atiyah from the 1950's. We prove also results on real line subbundles in higher genus using work of Lange-Narasimhan.