Extremal Kähler metrics on projective bundles over a curve

Let M = P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E -> Sigma over a compact complex curve Sigma of genus >= 2. Building on ideas of Fujiki (1992)[27], we prove that M admits a Kahler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kahler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kahler metrics in Kahler classes sufficiently far from the boundary of the Kahler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kahler manifolds with fibres isomorphic to a given tone Kahler variety. We discuss various ramifications of our approach to this class of manifolds.