A Hitchin-Kobayashi correspondence for Kaehler fibrations

Abstract

Let $X$ be a compact Kaehler manifold and $E\to X$ a principal $K$ bundle, where $K$ is a compact connected Lie group. Let ${\mathcal{A}}^{1,1}$ be the set of connections on $E$ whose curvature lies in ${\Omega }^{1,1}(E{\times }_{Ad}\mathfrak{k})$, where $\mathfrak{k}$ is the Lie algebra of $K$. Endow $\mathfrak{k}$ with a nondegenerate biinvariant bilinear pairing. This allows to identify $\{\frak k}\simeq{\frak k}^*$. Let $F$ be a Kaehler left $K$-manifold and suppose that there exists a moment map $\mu$ for the action of $K$ on $F$. Let $\mathcal{S}=\Gamma (E{\times }_{K}F)$. In this paper we study the equation $$\Lambda F_A+\mu (\Phi )=c$$ for $A\in {\mathcal{A}}^{1,1}$ and a section $\Phi \in \mathcal{S}$, where $c\in \mathfrak{k}$ is a fixed central element. We study which orbits of the action of the complex gauge group on ${calA}^{1,1}\times \mathcal{S}$ contain solutions of the equation, and we define a positive functional on ${calA}^{1,1}\times \mathcal{S}$ which generalises the Yang-Mills-Higgs functional and whose local minima coincide with the solutions of the equation.

41 pages, no figures, Latex2e