Vector bundles over a real elliptic curve

TL;DR

This paper classifies and describes the topology of moduli spaces of semi-stable and indecomposable vector bundles over a real elliptic curve, providing new insights into their structure and real algebraic properties.

Contribution

It determines the isomorphism classes of these moduli spaces and describes their topology and modular interpretation over real elliptic curves.

Findings

Classified moduli spaces of semi-stable vector bundles for coprime rank and degree.

Described the topology of the real locus of these moduli spaces.

Determined the isomorphism classes of moduli spaces of indecomposable bundles.

Abstract

Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r and d, we determine the isomorphism class of the moduli space of semi-stable vector bundles of rank r and degree d on the curve. When r and d are coprime, we describe the topology of the real locus and give a modular interpretation of its points. We also study, for arbitrary rank and degree, the moduli space of indecomposable vector bundles of rank r and degree d, and determine its isomorphism class as a real algebraic variety.