Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications

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