Representations of the fundamental group of a surface in PU(p,q) and holomorphic triples

Abstract

We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat bundles associated to the representations. Our results show that for each allowed value of these invariants, which are bounded by a Milnor-Wood type inequality, there is a unique non-empty connected component. Interpreting the moduli space of representations as a moduli space of Higgs bundles, we take a Morse theoretic approach using a certain smooth proper function on the Higgs moduli space. 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 non-empty and irreducible.

6 pages, to appear in C. R. Acad. Sci. Paris Ser. I Math

Cited by in corpus (7)

• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• Surface group representations in PU(p,q) and Higgs bundles
• Hitchin-Kobayashi correspondence, quivers, and vortices
• Surface group representations, Higgs bundles, and holomorphic triples
• Maximal Higgs bundles for adjoint forms via Cayley correspondence
• Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces
• Coherent systems and Brill-Noether theory