DIMENSIONAL REDUCTION, ${\rm SL} (2, {\mathbb C})$-EQUIVARIANT BUNDLES AND STABLE HOLOMORPHIC CHAINS

Abstract

In this paper we study gauge theory on [Formula: see text]-equivariant bundles over X × ℙ1, where X is a compact Kähler manifold, ℙ1 is the complex projective line, and the action of [Formula: see text] 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 ℰi and morphisms ℰi → ℰi-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.