The Moduli of Flat U(p, 1) Structures on Riemann Surfaces

For X a smooth projective curve over ℂ of genus g > 1, Hom+(π1(X), U(p, 1))/U(p, 1) is the moduli space of flat semi-simple U(p, 1)-connections on X. There is an integer invariant, τ, the Toledo invariant associated with each element in Hom+(π1(X), U(p, 1))/U(p, 1). This paper shows that Hom+(π1(X), U(p, 1))/U(p, 1) has one connected component corresponding to each τ & in 2 ℤ with −2(g−1) ≤τ≤2(g−1). Therefore the total number of connected components is 2(g−1)+1.