Curvature formulas and curvature strict positivity of direct image bundles

In this paper, we consider the curvature strict positivity of direct image bundles (vector bundles) associated to a strictly pseudoconvex family of bounded domains.The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded domains is strictly positive in the sense of Nakano even if the curvature of the original vector bundle is just Nakano positive. Based on our (I and my coauthors) previous results, this result further demonstrates that strictly pseudoconvex domains and pseudoconvex domains have very different geometric properties. To consider the curvature strict positivity, we will first construct a curvature formula for a direct image bundle, then the curvature strict positivity will be a simple consequence of it. As applications to convex analysis, we get a corresponding version of Prékopa's Theorem, i.e., we get the curvature strict positivity of a strictly convex family of bounded domains.In the last, we give a flatness criterion for the direct image bundles.onvex family of bounded domains.