Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds

Fusheng Deng; Jiafu Ning; Zhiwei Wang; Xiangyu Zhou

doi:10.5802/crmath.675

Research article - Dynamical systems

Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds

Fusheng Deng ¹; Jiafu Ning ² ; Zhiwei Wang ³; Xiangyu Zhou ^1,⁴

¹ School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China
² School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China
³ School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China
⁴ Institute of Mathematics, Academy of Mathematics and Systems Sciences and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences Beijing, 100190, P. R. China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1707-1715.

Abstract
Résumé

We show that a singular Hermitian metric on a holomorphic vector bundle over a Stein manifold which is negative in the sense of Griffiths (resp. Nakano) can be approximated by a sequence of smooth Hermitian metrics with the same curvature negativity. We also show that a smooth Hermitian metric on a holomorphic vector bundle over a Stein manifold restricted to a submanifold which is negative in the sense of Griffiths (resp. Nakano) can be extended to the whole bundle with the same curvature negativity.

Nous montrons qu’une métrique hermitienne singulière sur un fibré vectoriel holomorphe sur une variété de Stein qui est négative au sens de Griffiths (resp. Nakano) peut ê tre approximé par une séquence de métriques hermitiennes lisses avec la même négativité de courbure. Nous montrons également qu’une métrique hermitienne lisse sur un fibré vectoriel holomorphe sur une variété de Stein restreinte à une sous-variété ce qui est négatif au sens de Griffiths (resp. Nakano) peut être étendu à l’ensemble du faisceau avec la même négativité de courbure.

Received: 2023-09-21
Revised: 2024-02-24
Accepted: 2024-08-08
Published online: 2024-11-25

Zbl

DOI: 10.5802/crmath.675

Classification: 32Q28
Keywords: Approximation, extension, singular hermitian metric, Griffiths negative, Nakano negative
Mots-clés : Approximation, extension, métrique hermitienne singulière, négativité de Griffiths, négativité de Nakano

Author's affiliations:

Fusheng Deng ¹; Jiafu Ning ²; Zhiwei Wang ³; Xiangyu Zhou ^{1, 4}

¹ School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China
² School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China
³ School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China
⁴ Institute of Mathematics, Academy of Mathematics and Systems Sciences and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences Beijing, 100190, P. R. China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Fusheng Deng and Jiafu Ning and Zhiwei Wang and Xiangyu Zhou},
     title = {Approximation and extension of {Hermitian} metrics on holomorphic vector bundles over {Stein} manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1707--1715},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.675},
     zbl = {07949980},
     language = {en},
}

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Fusheng Deng; Jiafu Ning; Zhiwei Wang; Xiangyu Zhou. Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1707-1715. doi : 10.5802/crmath.675. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.675/

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