Vector bundles over a real elliptic curve

Indranil Biswas and Florent Schaffhauser

Vol. 283 (2016), No. 1, 43–62

DOI: 10.2140/pjm.2016.283.43

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 semistable 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.

Keywords

Elliptic curves, vector bundles on curves, real algebraic varieties

Mathematical Subject Classification 2010

Primary: 14H52, 14H60, 14P25

Milestones

Received: 21 September 2015

Revised: 28 December 2015

Accepted: 7 January 2016

Published: 14 June 2016