A note on the moduli spaces of holomorphic and logarithmic connections over a compact Riemann surface

TL;DR

This paper investigates the structure of moduli spaces of holomorphic and logarithmic connections on a compact Riemann surface, computing their Chow groups, global differential operators, and establishing Torelli-type results and rational connectedness.

Contribution

It provides new insights into the geometry and topology of these moduli spaces, including Chow groups, differential operators, Torelli theorems, and rational connectedness properties.

Findings

Chow groups of the moduli spaces are determined.

Global sections of differential operators are computed and shown to be constant under certain conditions.

A Torelli-type theorem for logarithmic connections is established.

Abstract

Let $X$ be a compact Riemann surface of genus $g \geq 3$. We consider the moduli space of holomorphic connections over $X$ and the moduli space of logarithmic connections singular over a finite subset of $X$ with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational connectedness of these moduli spaces.