Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type

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 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.