The Moduli of Flat Pu(2,1) Structures on Riemann Surfaces

For a compact Riemann surface X of genus g > 1, Hom(π1(X),PU(p, q))/PU(p, q) is the moduli space of flat PU(p, q)-connections on X. There are two integer invariants, dP , dQ, associated with each σ ∈ Hom(π1(X),PU(p, q))/ PU(p, q). These invariants are related to the Toledo invariant τ by τ = 2P −pdQ p+q . This paper shows, via the theory of Higgs bundles, that if q = 1, then −2(g − 1) ≤ τ ≤ 2(g − 1). Moreover, Hom(π1(X),PU(2, 1))/PU(2, 1) has one connected component corresponding to each τ ∈ 2 3 Z with −2(g − 1) ≤ τ ≤ 2(g − 1). Therefore the total number of connected components is 6(g − 1) + 1.