Curvature of vector bundles associated to holomorphic fibrations

Abstract

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 $X_y$ to $L\otimes K_{X/Y}$, 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.