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

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\mathcal{M}_{lc} (n,d)$ 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 $\mathfrak{gl} (n, \mathbb{C})$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\mathcal{M}^{\prime}_{lc} (n, d)$ and $\mathcal{M}_{lc} (n, L)$ be the moduli spaces parametrising all pairs $(E, D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^n E, \tilde{D}) \cong (L, D_L)$ respectively. Let $\mathcal{M}^{\prime}_{lc} (n, L) \subset \mathcal{M}_{lc} (n, L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\mathcal{M}^{\prime}_{lc} (n, d)$ and $\mathcal{M}^{\prime}_{lc} (n, L)$ and compute their Picard groups. We also show that $\mathcal{M}^{\prime}_{lc} (n, L)$ and hence $\mathcal{M}_{lc} (n, L)$ 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.

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Received 13 August 2019

Accepted 1 April 2020

Published 2 June 2021