We study adiabatic limits of Ricci-flat Kähler 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-Ampère 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.
"Adiabatic limits of Ricci-flat Kähler metrics." J. Differential Geom. 84 (2) 427 - 453, February 2010. https://doi.org/10.4310/jdg/1274707320