A Hitchin-Kobayashi correspondence for Kahler fibrations

Let X be a compact Kahler manifold and E --> X a principal K bundle, where k is a compact connected Lie group. Let f = Lie(K), and fix on f a nondegenerate biinvariant bilinear pairing. This allows to identify f similar or equal to f(*). Let F be a Kahler left K-manifold and suppose that there exists a moment map mu : F --> f* for the action of K on F. Let A(1,1) be the set of connections on E whose curvature lies in Ohm (1,1) (E x(Ad) f). Let Y = Gamma (E x(K) F). In this paper we study the equation LambdaF(A) + mu(Phi) = c for A is an element of A(1,1) on E and a section Phi is an element of Y, where F-A is the curvature of A and c is an element of f is a fixed central element. We study which orbits of the action of the complex gauge group on A(1,1) x Y' contain solutions of the equation and we define a positive functional on A(1,1) x Y' which generalises the Yang-Mills-Higgs functional and whose local minima coincide with the solutions of the equation.