Nakano positivity of singular Hermitian metrics and vanishing theorems \n of Demailly–Nadel–Nakano type

Abstract

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.

Bogomolov-Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics

“…For Nakano semi-positivity of singular Hermitian metrics, we already know one definition in [Ina22], which is based on the optimal L 2 -estimate condition in [HI20], [DNWZ20] and is equivalent to the usual definition for the smooth case. Definition 4.6.…”

“…In [Nad89], Nadel proved that I (h) is coherent by using the Hörmander L 2 -estimate. After that, as holomorphic vector bundles case, Hosono and Inayama proved that E (h) is coherent if h is Nakano semi-positive in the sense of singular as in Definition 4.6 in [HI20] and [Ina22].…”

“…This phenomenon is first mentioned in [Ina21] as an extension of the properties seen in plurisubharmonic functions. After that, it is extended to the case of Nakano semipositivity in [Ina22] and then to the case of the (p, q)-L 2 -estimate condition with p+q > n in [Wat21].…”

“…After that, Nakano semi-positivity for singular Hermitian metrics (see Definition 4.6) was defined in [Ina22], which establishes the singular-type Nakano vanishing theorem, i.e. the Demailly-Nadel type vanishing theorem extended to holomorphic vector bundles.…”

“…[Dem-book, ChapterVII, Corollary 9.4], [LSY13]) to singular Hermitian metrics and which can also be considered as a generalization of the Demailly-Nadel vanishing theorem and Theorem 1.1 to holomorphic vector bundles. Here, the generalization up to (n, q)-forms for singular Hermitian metrics is already known in [Ina22].…”