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

TL;DR

This paper studies the structure and properties of moduli spaces of logarithmic connections on vector bundles over a compact Riemann surface, including their compactifications, Picard groups, and algebraic functions.

Contribution

It introduces natural compactifications of these moduli spaces and computes their Picard groups, revealing their algebraic and holomorphic function properties.

Findings

Computed Picard groups of the moduli spaces.

Showed the absence of non-constant algebraic functions.

Established the existence of non-constant holomorphic functions.

Abstract

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ 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 $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, \tilde{D}) \cong (L,D_L)$ respectively. Let $\cat{M}'_{lc}(n,L) \subset \cat{M}_{lc}(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{lc}(n,d)$ and $\cat{M}'_{lc}(n,L)$ and compute their Picard groups. We also show that $\cat{M}'_{lc}(n,L)$ and hence $\cat{M}_{lc}(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.