On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case

Jiafu Ning; Zhiwei Wang; Xiangyu Zhou

doi:10.5802/afst.1767

Jiafu Ning ¹; Zhiwei Wang ²; Xiangyu Zhou ³

¹ School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China
² Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China
³ 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 and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 183-195.

Abstract
Abstract

In the present paper, we show that given a compact Kähler manifold $(X, ω)$ with a Kähler metric (not necessarily Hodge metric) $ω$ , and a complex submanifold $V \subset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$ , then any quasi-plurisubharmonic function $φ$ on $V$ such that ${ω |}_{V} + \sqrt{- 1} \partial \bar{\partial} {φ \geq ε ω |}_{V}$ with $ε > 0$ can be extended to a quasi-plurisubharmonic function $Φ$ on $X$ , such that $ω + \sqrt{- 1} \partial \bar{\partial} Φ \geq ε^{'} ω$ for some $ε^{'} > 0$ . This gives a partial answer to a question raised by Coman–Guedj–Zeriahi [4].

Dans le présent article, nous montrons que étant donné une variété compacte de kählérienne $(X, ω)$ avec une métrique de kählérienne (pas nécessairement une métrique de Hodge) $ω$ , et une sous-variété complexe $V \subset X$ de dimension positif, si $V$ a une structure de rétraction holomorphe dans $X$ , alors toute fonction quasi-plurisousharmonique $φ$ sur $V$ telle que ${ω |}_{V} + \sqrt{- 1} \partial \bar{\partial} {φ \geq ε ω |}_{V}$ avec $ε > 0$ peut être étendue é une fonction quasi-plurisousharmonique $Φ$ sur $X$ , telle que $ω + \sqrt{- 1} \partial \bar{\partial} Φ \geq ε^{'} ω$ pour quelques $ε^{'} > 0$ . Ceci donne une réponse partielle à une question soulevée par Coman–Guedj–Zeriahi [4].

Received: 2021-04-12
Accepted: 2022-03-15
Published online: 2024-07-26

DOI: 10.5802/afst.1767

Author's affiliations:

Jiafu Ning ¹; Zhiwei Wang ²; Xiangyu Zhou ³

¹ School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China
² Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China
³ 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 and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Jiafu Ning and Zhiwei Wang and Xiangyu Zhou},
     title = {On the extension of {K\"ahler} currents on compact {K\"ahler} manifolds: holomorphic retraction case},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {183--195},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Jiafu Ning; Zhiwei Wang; Xiangyu Zhou. On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 183-195. doi : 10.5802/afst.1767. https://afst.centre-mersenne.org/articles/10.5802/afst.1767/

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