Curvature strict positivity of direct image bundles associated to pseudoconvex families of domains

We consider the curvature strict positivity of the direct image bundle associated to a 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 circular domains or Reinhardt domains, with arbitrary plurisubharmonic weights, are strictly positive in the sense of Nakano. This result has some applications in complex analysis and convex analysis. We also investigate that the main result implies a remarkable result of Berndtsson which states that, for an ample vector bundle E over a compact complex manifold X and any \(k\ge 0\) $k\ge 0$ , the bundle \(S^kE\otimes \det E\) $S^{k}E\otimes detE$ admits a Hermitian metric whose curvature is strictly positive in the sense of Nakano, where \(S^kE\) $S^{k}E$ is the k-th symmetric product of E. The two main ingredients in the argument of the main theorems are Berndtsson’s estimate of curvature of direct image bundles and Deng–Ning–Wang–Zhou’s characterization of the curvature Nakano positivity of Hermitian vector bundles in terms of \(L^2\) $L^2$ -estimate of \({\bar{\partial }}\) $\overline{\partial }$ .