Moduli spaces of parabolic $U(p,q)$-Higgs bundles

Using the $L^2$-norm of the Higgs field as a Morse function, we count the number of connected components of 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. 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. We apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.