  Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we
show that the direct image of an adjoint semi-ample line bundle by a projective
submersion has a continuous metric with Griffiths semi-positive curvature. This
shows that for every holomorphic semi-ample vector bundle $E$ on a complex
manifold, and every positive integer $k$, the vector bundle $S^kE\otimes\det E$
has a continuous metric with Griffiths semi-positive curvature. If $E$ is ample
on a projective manifold, the metric can be made smooth and Griffiths positive.
