  This article finds constant scalar curvature Kahler metrics on certain
compact complex surfaces. The surfaces considered are those admitting a
holomorphic submersion to a curve, with fibres of genus at least 2. The proof
is via an adiabatic limit. An approximate solution is constructed out of the
hyperbolic metrics on the fibres and a large multiple of a certain metric on
the base. A parameter-dependent inverse function theorem is then used to
perturb the approximate solution to a genuine solution in the same cohomology
class. The arguments also apply to certain higher dimensional fibred Kahler
manifolds.
