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Principal Schottky bundles over Riemann surfaces

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Abstract

We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky G-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino (Manuscr Math 105:69–83, 2001) to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.

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Notes

  1. Interestingly, the consideration of the Schottky uniformization problem for vector bundles over Mumford curves, in the framework of p-adic analysis, has furnished stronger results. (see [11]).

  2. We are using a left action both on Y and on G; this was chosen (other options would be equivalent) for a standard use of Fox calculus in Sect. 8.

  3. For a general real Lie group, the analogous pairing defines a smooth (\(C^{\infty }\)) symplectic structure, see [20].

  4. Note that the case \(X=\mathbb {P}^{1}\) (\(g=0\)) is irrelevant, as \(\pi _{1}\) is trivial and so are Schottky representations.

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Acknowledgements

We thank I. Biswas, E. Franco, P. B. Gothen, C. Meneses-Torres and A. Oliveira for several useful discussions on Schottky bundles and related subjects, and the referees for clarifying comments. The last author thanks the organizers of the Simons Center for Geometry and Physics workshop on Higgs bundles, and L. Schaposnik and D. Baraglia for details on their construction of (A,B,A) branes.

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Authors and Affiliations

  1. Departamento Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

    A. C. Casimiro

  2. Escola Superior de Tecnologia e Gestão de Leiria, Campus 2, Morro do Lena, Alto do Vieiro, Apartado 4163, 2411-901, Leiria, Portugal

    S. Ferreira

  3. Departamento de Matemática, Faculdade de Ciências, Univ. de Lisboa, Edf. C6, Campo Grande, 1749-016, Lisbon, Portugal

    C. Florentino

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  1. A. C. Casimiro
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  2. S. Ferreira
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  3. C. Florentino
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Correspondence to C. Florentino.

Additional information

This work was partially supported by the Projects PTDC/MAT/120411/2010, PTDC/MAT-GEO/0675/2012 and EXCL/MAT-GEO/0222/2012, UID/MAT/00297/2013, FCT, Portugal, and by the USA NSF Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).

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Casimiro, A.C., Ferreira, S. & Florentino, C. Principal Schottky bundles over Riemann surfaces. Geom Dedicata 201, 379–409 (2019). https://doi.org/10.1007/s10711-018-0398-2

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