Curvature of higher direct images of sheaves of twisted holomorphic forms

TL;DR

This paper derives a general curvature formula for higher direct images of twisted holomorphic forms in Kähler families, analyzing special cases and applications to moduli spaces and Weil-Petersson metrics.

Contribution

It introduces a comprehensive curvature formula for these higher direct images and explores their properties in various geometric contexts, including flat and Hermite-Einstein bundles.

Findings

Derived a general curvature formula for higher direct images

Analyzed special cases with line bundles and specific indices

Applied results to moduli spaces and Weil-Petersson metrics

Abstract

This paper investigates the curvature properties of higher direct images $ R^qf_*\Omega_{X/S}^p(E)$, where $f: X\rightarrow S$ is a family of compact K\"ahler manifolds equipped with a hermitian vector bundle $E \rightarrow X$. We derive a general curvature formula and explore several special cases, including those where $p + q = n$, $q = 0$, and $p = n$, with $E$ being a line bundle. Furthermore, the paper examines the curvature in the context of fiberwise hermitian flat cases, families of Hermite-Einstein vector bundles, and applications to moduli spaces and Weil-Petersson metrics, providing some insight into their geometric and analytical properties.