Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Abstract

Let S be a finite subset of a compact connected Riemann surface X of genus g ≥ 2. Let M lc (n, d) denote the moduli space of pairs (E, D), where E is a holomorphic vector bundle over X and D is a logarithmic connection on E singular over S, with fixed residues in the centre of gl(n, C), where n and d are mutually corpime. Let L denote a fixed line bundle with a logarithmic connection D L singular over S. Let M ′ lc (n, d) and M lc (n, L) be the moduli spaces parametrising all pairs (E, D) such that underlying vector bundle E is stable and (be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of M ′ lc (n, d) and M ′ lc (n, L) and compute their Picard groups. We also show that M ′ lc (n, L) and hence M lc (n, L) do not have any non-constant algebraic functions but they admit non-constant holomorphic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over S, with arbitrary residues.

Line bundles on the moduli space of parabolic connections over a compact Riemann surface

“…In [15], the Picard group of the moduli space of parabolic vector bundles has been computed. Now using the techniques from [8], and [21], we show the following (see Theorem 3.3) Pic(M pc (r, d, α)) ∼ = Pic(M(r, d, α)).…”