Parabolic U(p,q)-Higgs bundles

In this thesis we study parabolic U(p, q)-Higgs bundles on a compact Riemann surface with a finite set of marked points. These objects correspond to representations of the fundamental group of the Riemann surface with punctures at the parabolic points in U(p, q), with fixed compact holonomy classes around the marked points. Our approach combines techniques used by Bradlow, Garcia-Prada and Gothen [BGG] in the non-parabolic case as well as those used by Garcia-Prada, Gothen and Munoz in [GGM] to study the topology of parabolic GL(3,C)-Higgs bundles. The strategy is to use the Bott–Morse theoretic techniques introduced by Hitchin in [H]. The connectedness properties of the moduli space reduces to the connectedness of a certain moduli space of parabolic triples introduced by Biquard and Garcia-Prada in [BG] in connection with the study of the parabolic vortex equations and instantons of infinite energy. Much of the work is devoted to a thorough study of these moduli spaces of triples and its connectedness properties. Our main results include the counting of number of connected components of the moduli space of parabolic U(n, 1)-Higgs bundles as well as the computation of the Poincare polynomial of the moduli space of parabolic U(2, 1)-Higgs bundles with one marked point.

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