  This paper is a sequel to \cite{Berndtsson}. In that paper we studied the
vector bundle associated to the direct image of the relative canonical bundle
of a smooth K\"ahler morphism, twisted with a semipositive line bundle. We
proved that the curvature of a such vector bundles is always semipositive (in
the sense of Nakano). Here we adress the question if the curvature is strictly
positive when the Kodaira-Spencer class does not vanish. We prove that this is
so provided the twisting line bundle is stricty positive along fibers, but not
in general.
