Hodge metrics and the curvature of higher direct images

TL;DR

This paper proves that higher direct images of the relative canonical bundle, when twisted by a semi-positive bundle and equipped with a Hodge metric, are locally free and semi-positively curved, extending curvature understanding.

Contribution

It introduces a new approach combining harmonic theory and curvature computation to establish local freeness and semi-positivity of higher direct images.

Findings

Higher direct images are locally free.

They possess semi-positive curvature.

The results extend previous curvature theorems.

Abstract

Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct images bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.