On the logarithmic connections over curves

TL;DR

This paper explores the structure of logarithmic connections on curves, establishing a correspondence with orbifold and parabolic connections, and analyzing fixed points under line bundle actions, revealing triviality in certain cases.

Contribution

It introduces a dictionary linking orbifold and parabolic logarithmic connections and characterizes fixed points under line bundle actions on the moduli space.

Findings

Established a correspondence between orbifold and parabolic connections.

Identified fixed points as push-forwards from Galois covers.

Proved cohomological triviality of line bundle actions in the coprime case.

Abstract

We study two different actions on the moduli spaces of logarithmic connections over smooth complex projective curves. Firstly, we establish a dictionary between logarithmic orbifold connections and parabolic logarithmic connections over the quotient curve. Secondly, we prove that fixed points on the moduli space of connections under the action of finite order line bundles are exactly the push-forward of logarithmic connections on a certain unramified Galois cover of the base curve. In the coprime case, this action of finite order line bundles on the moduli space is cohomologically trivial.