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The Quarterly Journal of Mathematics Advance Access published online on April 19, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han001
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

MODULI SPACES OF PARABOLIC U(p, q)-HIGGS BUNDLES

O. García-Prada{dagger}

Instituto de Ciencias Matemáticas CSIC-UAM-UCM-UC3M, Consejo Superior de Investigaciones Científicas, Serrano 121, 28006 Madrid, Spain

M. Logares {ddagger}

Departamento de Matematica Pura, Facultade de Ciencias, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Vicente Muñoz §

Instituto de Ciencias Matemáticas CSIC-UAM-UCM-UC3M, Consejo Superior de Investigaciones Científicas, Serrano 113 bis, 28006 Madrid, Spain
Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

{dagger} Corresponding author. E-mail: oscar.garcia-prada{at}uam.es

Received 15 March 2006; revised 21 January 2008
  Abstract

Using the L2-norm of the Higgs field as a Morse function, we study the moduli space of parabolic U(p, q)-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. When the parabolic degree is zero this space is homeomorphic to the moduli space of representations of the fundamental group of the punctured surface in U(p, q), with fixed compact holonomy classes around the marked points. By means of this homeomorphism we count the number of connected components of this moduli space of representations. Finally, we apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.

{ddagger} E-mail: mlogares{at}fc.up.pt

§ E-mail: vicente.munoz{at}imaff.cfmac.csic.es


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