Dimensional reduction, SL(2, C)-equivariant bundles and stable holomorphic chains

In this paper we study gauge theory on SL(2, C)-equivariant bundles over X x P-1, where X is a compact Kahler manifold, pi is the complex projective line, and the action of SL(2, C) is trivial on X and standard on pl. We first classify these bundles, showing that they are in correspondence with objects on X - that we call holomorphic chains - consisting of a finite number of holomorphic bundles epsilon (i) and morphisms epsilon --> epsilon (i) - 1 We then prove a Hitchin-Kobayashi correspondence relating the existence of solutions to certain natural gauge-theoretic equations and an appropriate notion of stability for an equivariant bundle and the corresponding chain. A central tool in this paper is a dimensional. reduction procedure which allow us to go from X x P-1 to X.