Narrow Results

We generalize the descriptions of vortex moduli spaces in [4] to more than one section with adiabatic constant $s$. The moduli space is topologically independent of $s$ but is not compact with respect to $C^{\infty }$ topology. Following [17], we construct a Gromov limit for vortices of fixed energy, and attempt to compactify the moduli space via bubble trees with possibly conical bubbles (or raindrops).

In this paper we study gauge theory on -equivariant bundles over X × ℙ¹, where X is a compact Kähler manifold, ℙ¹ is the complex projective line, and the action of 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 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 X × ℙ¹ to X.