Families of Calabi-Yau manifolds and canonical singularities

Given a polarized family of varieties over Δ, smooth over Δ^×and with smooth fibers Abstract, 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 Kähler metrics in the polarization class on the smooth fibers have uniformly bounded diameter, or are uniformly volume non-collapsed.