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Moduli spaces of vector bundles over a Klein surface

Abstract

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).

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Correspondence to Florent Schaffhauser.

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Dedicated to the memory of Paulette Libermann (1919–2007).

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Schaffhauser, F. Moduli spaces of vector bundles over a Klein surface. Geom Dedicata 151, 187–206 (2011). https://doi.org/10.1007/s10711-010-9526-3

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  • DOI: https://doi.org/10.1007/s10711-010-9526-3

Keywords

  • Moduli spaces
  • Lagrangian submanifolds
  • Klein surfaces

Mathematics Subject Classification (2000)

  • 53D30
  • 53D12
  • 30F50