Curvature of higher direct images of sheaves of twisted holomorphic forms

This paper investigates the curvature properties of higher direct images $R^q{f}_{\ast }{\mathrm{\Omega }}_{X\mathrm{/}S}^p(E)$, where $f:X\to S$ is a family of compact Kähler manifolds equipped with a hermitian vector bundle $E\to X$. 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.