Principal bundles over a real algebraic curve

2011/08/01 by Biswas, Indranil, Hurtubise, Jacques
#14H60 #14P99 #53C07 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1108.0234

Abstract

Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution σ. Let G be a connected complex reductive affine algebraic group, and let σ_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to σand σ_G. We prove that the points defined over \mathbb R of the smooth locus of a moduli space of principal G-bundles on X are precisely these objects, under the assumption that \rm genus(X) > 2. Stable, semistable and polystable bundles are defined in this context. Relationship between any of these properties and the corresponding property of the underlying holomorphic principal G-bundle is explored. A bijective correspondence between unitary representations and polystable objects is established.

Related

• Holomorphic principal bundles over elliptic curves
• Moduli space of principal sheaves over projective varieties
• Fundamental group of moduli of principal bundles on curves
• Pseudo-real principal G-bundles over a real curve
• Projective moduli space of semistable principal sheaves for a reductive group