Positivity of Singular Hermitian Metrics for Holomorphic Vector Bundles

2023/08/25 by Varolin, Dror
#32 #53 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.13364

Abstract

We introduce a notion of Nakano and Demailly positivity for singular Hermitian metrics of holomorphic vector bundles. Our definitions support the usual Hörmander and Nadel type vanishing theorems with estimates, at least on essentially Stein manifolds. As an application, we establish a sharp Ohsawa-Takegoshi type L² extension theorem. We use the method of Berndtsson and Lempert to prove the latter theorem, and for this purpose we require a Berndtsson-type positivity theorem for holomorphic vector bundles, which we also prove.

Related

• Singular hermitian metrics on vector bundles
• Nakano positivity of singular Hermitian metrics: Approximations and applications
• Optimal L² extension for holomorphic vector bundles with singular hermitian metrics
• The harmonic theory on a vector bundle with singular Hermitian metrics and positivity
• Bogomolov-Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics