Curvature of vector bundles associated to holomorphic fibrations

Let L be a (semi)-positive line bundle over a Kähler manifold, X, fibered over a complex manifold Y. Assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle E over Y whose fibers over points y are the spaces of global sections over Xy to L ⊗ Kx/y, endowed with the L2-metric, is (semi)-positive in the sense of Nakano. We also discuss various applications, among them a partial result on a conjecture of Griffiths on the positivity of ample bundles