Monodromy of a Class of Logarithmic Connections on an Elliptic Curve

TL;DR

This paper explicitly parametrizes certain logarithmic connections on elliptic curves, analyzes their monodromy and Galois groups, and explores the stability of associated vector bundles, advancing understanding of their geometric and algebraic properties.

Contribution

It provides an explicit parametrization and detailed analysis of monodromy, Galois groups, and stability for a class of logarithmic connections on elliptic curves.

Findings

Explicit parametrization of logarithmic connections

Determination of monodromy and differential Galois groups

Analysis of stability of underlying vector bundles

Abstract

The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their monodromy, differential Galois group and the underlying rank-2 vector bundle. The latter is described in terms of elementary transforms. The question of its (semi)-stability is addressed.