  We study the Ginzburg-Landau equations on Riemann surfaces of arbitrary
genus. In particular: - we construct explicitly the (local moduli space of
gauge-equivalent) solutions in a neighbourhood of the constant curvature ones;
- classify holomorphic structures on line bundles arising as solutions to the
equations in terms of the degree, the Abel-Jacobi map, and symmetric products
of the surface; - determine the form of the energy and identify when it is
below the energy of the constant curvature (normal) solutions.
