Abelian Vortices with Singularities

Understand

Let L --> X be a complex line bundle over a compact connected Riemann surface.

• We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure.
• These conditions appear naturally in the study of vortex configurations with constraints, or configurations invariant under the action of a finite group.
• We first show that the moduli space of singular vortex solutions is the same as in the regular case.