Families of Calabi-Yau manifolds and canonical singularities

Abstract

Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canonical singularities, answering a question of C.-L. Wang. This condition also implies that the Ricci-flat Kahler metrics in the polarization class on the smooth fibers have uniformly bounded diameter, or are uniformly volume non-collapsed.