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

Abstract

In this paper we study gauge theory on SL(2, ℂ)-equivariant bundles over X × ℙ1, where X is a compact Kähler manifold, ℙ1 is the complex projective line, and the action of SL(2, ℂ) is trivial on X and standard on ℙ1. 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 εi and morphisms εi → εi-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 × ℙ1 to X.