Generically surjective morphisms of holomorphic vector bundles via degenerations

We prove an $L^2$ theorem on generically surjective morphism of holomorphic vector bundles via a degeneration argument, generalizing the author's previous work on the $L^2$ division theorem of Skoda. The proof is based on Berndtsson's theorem on the positivity of direct image bundles and is inspired by Berndtsson and Lempert's proof of the $L^2$ extension theorem.