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

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 semistable 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 et al., and we use this to study the $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$-action $[\mathcal{E}] \mapsto [\overline{\sigma^*E}]$ on moduli varieties of stable holomorphic bundles on a complex curve with given real structure $\sigma$. 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 $\sigma$, we give an exact count of the number of connected components, thus generalizing to rank $r \gt 1$ the results of Gross and Harris on the Picard scheme of a real algebraic curve.