Surface group representations, Higgs bundles, and holomorphic triples
Abstract
Using the $L^2$ norm of the Higgs field as a Morse function, we study the moduli spaces of $U(p,q)$Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on $(p,q)$. A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. We prove that these moduli spaces of triples are irreducible and nonempty. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple $PU(p,q)$representations. The topological invariants of the flat bundles bundle are used to label components. These invariants are bounded by a MilnorWood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding component is nonempty and connected. If the coprimality condition does not hold, our results apply to the irreducible representations.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2002
 arXiv:
 arXiv:math/0206012
 Bibcode:
 2002math......6012B
 Keywords:

 Algebraic Geometry;
 14H30;
 14H60;
 32L05
 EPrint:
 106 pages