Curvature of vector bundles associated to holomorphic fibrations

Let $L$ be a (semi)-positive line bundle over a Kahler manifold, $X$, fibered over a complex manifold $Y$. Assuming the fibers are compact and non-singular we prove that the hermitian vector bundle $E$ over $Y$ whose fibers over points $y$ are the spaces of global sections over $X_y$ to $L\text{gr}{K}_{X\mathrm{/}Y}$ endowed with the $L^2$-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. This is a revised and much expanded version of a previous preprint with the title `` Bergman kernels and the curvature of vector bundles’'.