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January 2017 On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles
Florent Schaffhauser
J. Differential Geom. 105(1): 119-162 (January 2017). DOI: 10.4310/jdg/1483655861


Let $(M,\sigma)$ be a compact Klein surface of genus $g \geq 2$ and let $E$ be a smooth Hermitian vector bundle on $M$. Let $\tau$ be a Real or Quaternionic structure on $E$ and denote respectively by $\mathcal{G}^{\tau}_{\mathbb{C}}$ and $\mathcal{G}^{\tau}_{E}$ the groups of complex linear and unitary automorphisms of $E$ that commute to $\tau$. In this paper, we study the action of $\mathcal{G}^{\tau}_{\mathbb{C}}$ on the space $\mathcal{A}^{\tau}_{E}$ of $\tau$-compatible unitary connections on $E$ and show that the closure of a semi-stable $\mathcal{G}^{\tau}_{\mathbb{C}}$-orbit contains a unique $\mathcal{G}^{\tau}_{E}$-orbit of projectively flat connections. We then use this invariant-theoretic perspective to prove a version of the Narasimhan–Seshadri correspondence in this context: $S$-equivalence classes of semi-stable Real and Quaternionic vector bundes are in bijective correspondence with equivalence classes of certain appropriate representations of orbifold fundamental groups of Real Seifert manifolds over the Klein surface $(M,\sigma)$.


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Florent Schaffhauser. "On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles." J. Differential Geom. 105 (1) 119 - 162, January 2017.


Received: 11 October 2013; Published: January 2017
First available in Project Euclid: 5 January 2017

zbMATH: 1360.30037
MathSciNet: MR3592696
Digital Object Identifier: 10.4310/jdg/1483655861

Rights: Copyright © 2017 Lehigh University

Vol.105 • No. 1 • January 2017
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