## Journal of Symplectic Geometry

- J. Symplectic Geom.
- Volume 10, Number 4 (2012), 503-534.

### 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.

#### Article information

**Source**

J. Symplectic Geom., Volume 10, Number 4 (2012), 503-534.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jsg/1357153427

**Mathematical Reviews number (MathSciNet)**

MR2982021

**Zentralblatt MATH identifier**

06141803

#### Citation

Schaffhauser, Florent. Real points of coarse moduli schemes of vector bundles on a real algebraic curve. J. Symplectic Geom. 10 (2012), no. 4, 503--534. https://projecteuclid.org/euclid.jsg/1357153427