Extension of twisted Hodge metrics for Kähler morphisms

Abstract

Let f : X - Y be a holomorphic map of complex manifolds, which is proper, Khler, 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 R q f * (K X/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(R q f * (K X/Y E)) "over Y \ Z" as a singular Hermitian metric with semi-positive curvature "over Y ". As a particular consequence, if Y is projective, R q f * (K X/Y E) is weakly positive over Y \ Z in the sense of Viehweg.