Extremal K\"ahler 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 $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$ admits a K\"ahler metric of constant scalar curvature if and only if $E$ is polystable. We also address the more general existence problem of extremal K\"ahler 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 K\"ahler metrics in sufficiently small K\"ahler classes. The methods used to prove the above results apply to a wider class of manifolds, called {\it rigid toric bundles over a semisimple base}, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature K\"ahler manifolds with fibres isomorphic to a given toric K\"ahler variety. We discuss various ramifications of our approach to this class of manifolds.