Curvature of vector bundles associated to holomorphic fibrations

Abstract

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 $L\gr K_{X/Y}$ 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. This is a revised and much expanded version of a previous preprint with the title `` Bergman kernels and the curvature of vector bundles''.

This revision simplifies some proofs. An incorrect proof from the appendix has also been withdrawn (it was not used in the rest of the paper)

References in corpus (1)

• Variation of Bergman kernels of adjoint line bundles

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• Variation of Bergman kernels of adjoint line bundles
• Canonical singular hermitian metrics on relative logcanonical bundles
• A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
• Canonical measures and the dynamical systems of Bergman kernels
• Strict and non strict positivity of direct image bundles
• Ricci iterations and canonical Kähler-Einstein currents on log canonical pairs