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.
We study conditions of Hörmander's L 2 -estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of Hörmander-type condition, we show a converse of Hörmander L 2 -estimate under some regularity assumptions on an n-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson. We also define new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.
In this article, using a twisted version of Hörmander's L 2 -estimate, we give new characterizations of notions of partial positivity, which are uniform q-positivity and RC-positivity. We also discuss the definition of uniform q-positivity for singular Hermitian metrics.
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