Line bundles over a moduli space of logarithmic connections on a Riemann surface

Abstract.

We consider logarithmic connections, on rank n and degree d vector bundles over a compact Riemann surface X, singular over a fixed point x₀ ∈ X with residue in the center of ${\text{gl}}(n,\mathbb{C});$ the integers n and d are assumed to be mutually coprime. A necessary and sufficient condition is given for a vector bundle to admit such a logarithmic connection. We also compute the Picard group of the moduli space of all such logarithmic connections. Let $\mathcal{N}_D (L)$ denote the moduli space of all such logarithmic connections, with the underlying vector bundle being of fixed determinant L, and inducing a fixed logarithmic connection on the determinant line L. Let $\mathcal{N}'_.D (L) \subset \mathcal{N}_D (L)$ be the Zariski open dense subset parametrizing all connections such that the underlying vector bundle is stable. The space of all global sections of certain line bundles on $\mathcal{N}'_D (L)$ are computed. In particular, there are no nonconstant algebraic functions on $\mathcal{N}'_D (L).$ Therefore, there are no nonconstant algebraic functions on $\mathcal{N}_D (L),$ although $\mathcal{N}_D (L)$ is biholomorphic to a representation space which admits nonconstant algebraic functions. The moduli space $\mathcal{N}'_D (L)$ admits a natural compactification by a smooth divisor. We investigate numerically effectiveness of this divisor at infinity. It turns out that the divisor is not numerically effective in general.