  We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau
manifold which is the total space of a holomorphic fibration when the volume of
the fibers goes to zero. By establishing some new a priori estimates for the
relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics
collapse (away from the singular fibers) to a metric on the base of the
fibration. This metric has Ricci curvature equal to a Weil-Petersson metric
that measures the variation of complex structure of the Calabi-Yau fibers. This
generalizes results of Gross-Wilson for K3 surfaces to higher dimensions.
