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

الأصل · EN

For a compact Riemann surface X of genus g > 1, (π₁(X), U(p,1))/U(p,1) is the moduli space of flat (p,1)-connections on X. There is an integer invariant, τ, the Toledo invariant associated with each element in (π₁(X), U(p,1))/U(p,1). If q = 1, then -2(g-1) ≤ τ≤ 2(g-1). This paper shows that (π₁(X), U(p,1))/U(p,1) has one connected component corresponding to each τ∈ 2Z with -2(g-1) ≤ τ≤ 2(g-1). Therefore the total number of connected components is 2(g-1) + 1.