Journal of Differential Geometry

On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles

Florent Schaffhauser

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Abstract

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 $(M,\sigma)$.

Article information

Source
J. Differential Geom., Volume 105, Number 1 (2017), 119-162.

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

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1483655861

Digital Object Identifier
doi:10.4310/jdg/1483655861

Mathematical Reviews number (MathSciNet)
MR3592696

Zentralblatt MATH identifier
1360.30037

Citation

Schaffhauser, Florent. On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles. J. Differential Geom. 105 (2017), no. 1, 119--162. doi:10.4310/jdg/1483655861. https://projecteuclid.org/euclid.jdg/1483655861


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