ABSTRACT
In the short time since their discovery, the Seiberg-Witten equations have already proved to be a powerful tool in the study of smooth four-manifolds. Virtually all the hard-won gains that have been obtained using the heavy machinery of Donaldson invariants, can be recovered with a fraction of the effort if the anti-self-duality equations are replaced by the Seiberg-Witten equations. The impressive success of the original equations has naturally led to speculation about possible generalizations and other related sets of equations. The original equations as proposed by Seiberg-Witten are associated with a Hermitian line bundle, and thus with the Abelian group U. The main idea in our point of view is to exploit the special form of the Seiberg-Witten equations in the case where the four-manifold is a Kahler surface. In this case the original Seiberg-Witten equations are known to reduce essentially to familiar equations in gauge theory known as the Abelian vortex equations.
