Extension of twisted Hodge metrics for K\"ahler morphisms

Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over Y-Z'' as a singular Hermitian metric with semi-positive curvature ``over Y''. As a particular consequence, if Y is projective, R^qf_*(K_{X/Y} \otimes E) is weakly positive over Y-Z in the sense of Viehweg.