A Hitchin-kobayashi Correspondence for Kaehler Fibrations

Let X be a compact Kaehler manifold and E → X a principalK bundle, where K is a compact connected Lie group. Let A 1,1 be the set of connections on E whose curvature lies in Ω(E ×Ad k). Let k = Lie(K), and fix on k a nondegenerate biinvariant bilinear pairing. This allows to identify k ≃ k∗. Let F be a Kaehler left K-manifold and suppose that there exists a moment map μ : F → k∗ for the action of K on F . Let S = Γ(E ×K F ). In this paper we study the equation ΛFA + μ(Φ) = c for A ∈ A 1,1 on E and a section Φ ∈ S , where c ∈ k is a fixed central element. We study which orbits of the action of the complex gauge group on A 1,1 × S contain solutions of the equation and we define a positive functional on A 1,1 × S which generalises the Yang-Mills-Higgs functional and whose local minima coincide with the solutions of the equation.