Let f : X .仺 Y be a holomorphic map of complex manifolds,
which is proper, K丯ahler, 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. In
our previous paper, we discussed the Nakano semi-positivity of
Rqf(KX/Y . E) for q . 0 with respect to the so-called Hodge
metric, when the map f is smooth. Here we discuss the extension
of the induced metric on the tautological line bundle O(1) on the
projective space bundle P(Rqf(KX/Y . E)) 乬over Y \ Z乭 as a
singular Hermitian metric with semi-positive curvature 乬over Y 乭.
As a particular consequence, if Y is projective, Rqf(KX/Y . E)
is weakly positive over Y \ Z in the sense of Viehweg.