Optimal $L^2$ Extensions of Openness Type and Related Topics

Wang Xu; Xiangyu Zhou

doi:10.5802/crmath.437

Complex analysis and geometry

Optimal

L^{2}

Extensions of Openness Type and Related Topics

Wang Xu ¹ ; Xiangyu Zhou ²

¹ School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China
² Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 679-683.

Abstract
Résumé

We establish several optimal $L^{2}$ extension theorems of openness type on weakly pseudoconvex Kähler manifolds. We prove a product property for certain minimal $L^{2}$ extensions, which generalizes the product property of Bergman kernels. We describe a different approach to the Suita conjecture and its generalizations, which is based on a log-concavity for certain minimal $L^{2}$ integrals.

Nous établissons quelques théorèmes d’extension optimaux $L^{2}$ pour les formes ouvertes sur les variété Kähler faiblement pseudoconvexes. Nous prouvons les propriétés de produit de certaines extensions minimales de $L^{2}$ , qui généralisent les propriétés de produit du noyau Bergman. Sur la base de la concavité logarithmique de certaines intégrales minimales de $L^{2}$ , nous donnons une méthode différente pour la conjecture de Suita et son extension.

Received: 2022-02-05
Accepted: 2022-10-16
Published online: 2023-03-02

DOI: 10.5802/crmath.437

Classification: 32D15, 32A36, 32L05, 32W05, 32E10, 14C30, 30C40

Author's affiliations:

Wang Xu ¹; Xiangyu Zhou ²

¹ School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China
² Institute of Mathematics, Academy of Mathematics and Systems Science, 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 = {Wang Xu and Xiangyu Zhou},
     title = {Optimal $L^2$ {Extensions} of {Openness} {Type} and {Related} {Topics}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {679--683},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.437},
     language = {en},
}

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Wang Xu; Xiangyu Zhou. Optimal $L^2$ Extensions of Openness Type and Related Topics. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 679-683. doi : 10.5802/crmath.437. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.437/

References
Cited by

[1] Zbigniew Błocki Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math., Volume 193 (2013) no. 1, pp. 149-158 | DOI | MR | Zbl

[2] Zbigniew Błocki A lower bound for the Bergman kernel and the Bourgain-Milman inequality, Geometric aspects of functional analysis, Springer, 2014, pp. 53-63 | DOI | MR | Zbl

[3] Zbigniew Błocki; Włodzimierz Zwonek One dimensional estimates for the Bergman kernel and logarithmic capacity, Proc. Am. Math. Soc., Volume 146 (2018) no. 6, pp. 2489-2495 | DOI | MR | Zbl

[4] Jean-Pierre Demailly Estimations $L^{2}$ pour l’opérateur $\bar{\partial}$ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982) no. 3, pp. 457-511 | DOI | Numdam | MR | Zbl

[5] Jean-Pierre Demailly Scindage holomorphe d’un morphisme de fibrés vectoriels semi-positifs avec estimations $L^{2}$ , Seminar Pierre Lelong-Henri Skoda (Analysis), 1980/1981, Springer, 1982, pp. 77-107 | DOI | MR | Zbl

[6] Jean-Pierre Demailly Complex analytic and differential geometry (2012) (e-book)

[7] Jean-Pierre Demailly; Thomas Peternell; Michael Schneider Pseudo-effective line bundles on compact Kähler manifolds, Int. J. Math., Volume 12 (2001) no. 6, pp. 689-741 | DOI | MR | Zbl

[8] Qi’an Guan A sharp effectiveness result of Demailly’s strong openness conjecture, Adv. Math., Volume 348 (2019), pp. 51-80 | DOI | MR | Zbl

[9] Qi’an Guan; Zhitong Mi Concavity of minimal $L^{2}$ integrals releted to multiplier ideal sheaves, Peking Math. J. (2022) (https://doi.org/10.1007/s42543-021-00047-5) | Zbl

[10] Qi’an Guan; Zhitong Mi; Zheng Yuan Concavity property of minimal $L^{2}$ integrals with Lebesgue measurable gain II (2022) | arXiv

[11] Qi’an Guan; Xiangyu Zhou Optimal constant problem in the $L^{2}$ extension theorem, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 15-16, pp. 753-756 | DOI | Numdam | MR | Zbl

[12] Qi’an Guan; Xiangyu Zhou Optimal constant in an $L^{2}$ extension problem and a proof of a conjecture of Ohsawa, Sci. China Math., Volume 58 (2015) no. 1, pp. 35-59 | DOI | MR | Zbl

[13] Qi’an Guan; Xiangyu Zhou A solution of an $L^{2}$ extension problem with an optimal estimate and applications, Ann. Math., Volume 181 (2015) no. 3, pp. 1139-1208 | DOI | MR | Zbl

[14] Genki Hosono On sharper estimates of Ohsawa-Takegoshi $L^{2}$ -extension theorem, J. Math. Soc. Japan, Volume 71 (2019) no. 3, pp. 909-914 | MR | Zbl

[15] B. Jennane Extension d’une fonction définie sur une sous-variété avec contrôle de la croissance, Séminaire Pierre Lelong-Henri Skoda (Analyse), Année 1976/77, Springer, 1978, pp. 126-133 | DOI | MR | Zbl

[16] Nobuyuki Suita Capacities and kernels on Riemann surfaces, Arch. Ration. Mech. Anal., Volume 46 (1972), pp. 212-217 | DOI | MR | Zbl

[17] Wang Xu; Xiangyu Zhou Optimal $L^{2}$ extensions of openness type (2022) | arXiv

[18] Xiangyu Zhou; Langfeng Zhu An optimal $L^{2}$ extension theorem on weakly pseudoconvex Kähler manifolds, J. Differ. Geom., Volume 110 (2018) no. 1, pp. 135-186 | DOI | MR | Zbl

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