INVARIANT CONNECTIONS AND VORTICES

We study the vortex equations on a line bundle over a compact Kahler manifold. These are a generalization of the classical vortex equations over R2. We first prove an invariant version of the theorem of Donaldson, Uhlenbeck and Yau relating the existence of a Hermitian-Yang-Mills metric on a holomorphic bundle to the stability of such a bundle. We then show that the vortex equations are a dimensional reduction of the Hermitian-Yang-Mills equation. Using this fact and the theorem above we give a new existence proof for the vortex equations and describe the moduli space of solutions.