On a system of Seiberg-Witten equations

We introduce and study a system of Seiberg-Witten equations. These are r copies of the usual Seiberg-Witten equations coupled to each other involving r connections on r SpinC structures as well as r positive spinors and are Abelian generalizations of the Seiberg-Witten equations. For r=2, we show that the moduli space of solutions is a compact, orientable and smooth manifold. For minimal surfaces of general type, we are able to determine the basic classes.