  Given a proper, open, holomorphic map of K\"ahler manifolds, whose general
fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics
induce a hermitian metric on the relative canonical bundle over the regular
locus of the family. We show that the curvature form extends as a closed
positive current. Consequently the Weil-Petersson metric extends as a positive
current. In the projective case, the Weil-Petersson form is known to be the
curvature of a certain determinant line bundle, equipped with a Quillen metric.
As an application we get that after blowing up the singular locus, the
determinant line bundle extends, and the Quillen metric extends as singular
hermitian metric, whose curvature is a positive current.
