Real points of coarse moduli schemes of vector bundles on a real algebraic curve

TL;DR

This paper studies the real points of moduli spaces of vector bundles on real algebraic curves, providing a gauge-theoretic construction, relating to previous work, and establishing bounds and exact counts for connected components of fixed points under Galois action.

Contribution

It introduces a gauge-theoretic approach to moduli spaces of real and quaternionic vector bundles, and generalizes known results on the topology of real algebraic curves to higher rank vector bundles.

Findings

Connected components of fixed points are bounded by 2^g + 1.

Exact counts of connected components are provided considering all topological invariants.

The work generalizes results from rank 1 to higher rank vector bundles.

Abstract

We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semi-stable such bundles with fixed topological type. These spaces embed onto connected subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise (arxiv:0901.3071), and we use this to study the Galois action induced on moduli varieties of stable holomorphic bundles on a complex curve by a given real structure on the curve. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of that action by $2^g +1$, where $g$ is the genus of the curve. In fact, taking into account all the topological invariants of the real structure, we give an exact count of the number of connected components, thus generalising to rank $r > 1$ the results of Gross and Harris on the Picard scheme of a real algebraic curve.