  In this article, we propose a definition of Nakano semi-positivity of
singular Hermitian metrics on holomorphic vector bundles. By using this
positivity notion, we establish $L^2$-estimates for holomorphic vector bundles
with Nakano positive singular Hermitian metrics. We also show vanishing
theorems, which generalize both Nakano type and Demailly-Nadel type vanishing
theorems. As applications, we specifically construct globally Nakano
semi-positive singular Hermitian metrics for several bundles, and prove
vanishing theorems associated with them.
