  In this article we are interested in the differential geometric properties of
certain higher direct images of exterior powers of the sheaf of relative
differentials twisted with a line bundle. We obtain explicit curvature
formulas, especially in case where the said line bundle satisfies a natural
curvature assumption. Several applications are obtained, including a proof of a
result by Viehweg-Zuo in the context of a canonically polarized family of
maximal variation.
