L2L^2 estimates and vanishing theorems for holomorphic vector bundles equipped with singular Hermitian metrics

We investigate singular Hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ esitimates and vanishing theorems usually require an assumption that vector bundles are Nakano positive. However there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theromes by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.Comment: 19 page

Pseudonorms on direct images of pluricanonical bundles

We study pseudonorms on pluricanonical bundles over Stein manifolds. We prove that the pseudonorms determine holomorphic structures of Stein manifolds under certain assumptions. This theorem is a generalization of the result obtained by Deng, Wang, Zhang, and Zhou for bounded domains in $\mathbb{C}^n$. We also investigate Stein morphisms and the pseudonorms on direct images of pluricanonical bundles. Our main goal in this paper is to show that the pseudonorms also determine holomorphic structures of Stein morphisms. One important technique is an $L^{2/m}$-variant of the Ohsawa-Takegoshi extension theorem.Comment: 18 page

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 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.Comment: 28pages, v3: major revision. Theorem 1.7, 1.8, Subsection 2.3 and Section 6 were adde

From Hörmander’s L2L^2-estimates to partial positivity

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

From Hörmander’s L2L^2-estimates to partial positivity

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