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Moduli spaces of holomorphic triples over compact Riemann surfaces

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Abstract.

A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study non-emptiness, irreducibility, smoothness, and birational descriptions of these moduli spaces for a certain range of the parameter. Our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary Lie groups of indefinite signature ([5, 7]). Another application, that we study in this paper, is to the existence of stable bundles on the product of the surface by the complex projective line.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Steven B. Bradlow

  2. Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano, 113 bis, 28006, Madrid, Spain

    Oscar Gar-cía-Prada

  3. Departamento de Matemática Pura, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Peter B. Gothen

Authors
  1. Steven B. Bradlow
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  2. Oscar Gar-cía-Prada
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  3. Peter B. Gothen
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Corresponding author

Correspondence to Steven B. Bradlow.

Additional information

Members of VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER (EC FP5 Contract no.\ HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no.\ HPRN-CT-2000-00101).}

Partially supported by the National Science Foundation under grant DMS-0072073.

Partially supported by the Ministerio de Ciencia y Tecnología (Spain) under grant BFM2000-0024.

Partially supported by the Fundação para a Ciência e a Tecnologia (Portugal) through the Centro de Matemática da Universidade do Porto and through grant no.\ SFRH/BPD/1606/2000.

Partially supported by the Portugal/Spain bilateral Programme Acciones Integradas, grant nos.\ HP2000-0015 and AI-01/24.

Partially supported by a British EPSRC grant (October-December 2001).

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Bradlow, S., Gar-cía-Prada, O. & Gothen, P. Moduli spaces of holomorphic triples over compact Riemann surfaces. Math. Ann. 328, 299–351 (2004). https://doi.org/10.1007/s00208-003-0484-z

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  • DOI: https://doi.org/10.1007/s00208-003-0484-z

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