Positivity of Singular Hermitian Metrics for Holomorphic Vector Bundles

We introduce a notion of Nakano and Demailly positivity for singular Hermitian metrics of holomorphic vector bundles. Our definitions support the usual Hörmander and Nadel type vanishing theorems with estimates, at least on essentially Stein manifolds. As an application, we establish a sharp Ohsawa-Takegoshi type $L^2$ extension theorem. We use the method of Berndtsson and Lempert to prove the latter theorem, and for this purpose we require a Berndtsson-type positivity theorem for holomorphic vector bundles, which we also prove.