Monodromy of a Class of Logarithmic Connections on an Elliptic Curve

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.

Keywords

• elliptic curve
• ramified covering
• logarithmic connection
• bielliptic curve
• genus-2 curve
• monodromy
• Riemann-Hilbert problem
• differential Galois group
• elementary transformation
• stable bundle
• vector bundle