  We prove that the $L^2$ metric on the direct image of an adjoint positive
line bundle by a locally trivial submersion between projective manifolds is
Nakano positive, under the assumption that the typical fiber has zero first
Betti number. As a consequence, we get that the symmetric powers of an ample
vector bundle tensorized by its determinant are Nakano positive, in particular
Griffiths positive. This in turn gives vanishing theorems and an analytic
characterization of ample vector bundles.
