Characterizations of curvature positivity of Riemannian vector bundles and convexity or pseudoconvexity of bounded domains in [formula omitted] or [formula omitted] in terms of L2-estimate of d or [formula omitted] equation

Abstract

The first main result is a characterization of Nakano positivity of Riemannian vector bundles over bounded domains in terms of solvability of the d-equation with certain good L2 estimate condition. As an application, we give an alternative proof of the matrix-valued Prekopa's theorem that is originally proved by Raufi. The second main result contains a characterization of convexity of smoothly bounded domains in Rn in terms L2 estimate condition for the d-equation, and a characterization of pseudoconvexity of smoothly bounded domains in Cn in terms L2 estimate condition for the ∂¯-equation. Our methods are inspired by the recent works of Deng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles and positivity of direct image sheaves associated to holomorphic fibrations.