  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.
