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December 2012 Real points of coarse moduli schemes of vector bundles on a real algebraic curve
Florent Schaffhauser
J. Symplectic Geom. 10(4): 503-534 (December 2012).

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.

Citation

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Florent Schaffhauser. "Real points of coarse moduli schemes of vector bundles on a real algebraic curve." J. Symplectic Geom. 10 (4) 503 - 534, December 2012.

Information

Published: December 2012
First available in Project Euclid: 2 January 2013

zbMATH: 06141803
MathSciNet: MR2982021

Rights: Copyright © 2012 International Press of Boston

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Vol.10 • No. 4 • December 2012
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