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

TL;DR

This paper classifies SL(2,C)-equivariant bundles over product spaces involving a Kahler manifold and P^1, establishing a correspondence with holomorphic chains and linking gauge solutions to stability conditions via dimensional reduction.

Contribution

It introduces a classification of equivariant bundles as holomorphic chains and proves a Hitchin-Kobayashi correspondence using dimensional reduction techniques.

Findings

Equivariant bundles correspond to holomorphic chains on X.

Existence of gauge solutions relates to stability of chains.

Dimensional reduction simplifies the gauge theory analysis.

Abstract

In this paper we study gauge theory on SL(2,C)-equivariant bundles over XxP^1, where X is a compact Kahler manifold, P^1 is the complex projective line, and the action of SL(2,C) is trivial on X and standard on P^1. 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 E_i and morphisms E_i->E_{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 XxP^1 to X.