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 non-singular we prove that the hermitian vector bundle E whose fibers are the space of global sections to L⊗KX/Y endowed with the L-metric is (semi)-positive in the sense of Nakano. As an application we prove a partial result on a conjecture of Griffiths on the positivity of ample bundles. This is a revised and much expanded version of a previous preprint with the title “ Bergman kernels and the curvature of vector bundles”.