On the Ohsawa–Takegoshi $ {L}^{2}$ extension theorem and the twisted Bochner–Kodaira identity

Qiʼan Guan; Xiangyu Zhou; Langfeng Zhu

doi:10.1016/j.crma.2011.06.001

Analytic Geometry

On the Ohsawa–Takegoshi

L^{2}

extension theorem and the twisted Bochner–Kodaira identity
[Sur le théorème dʼextension

L^{2}

de Ohsawa–Takegoshi et lʼidentité tordue de Bochner–Kodaira]

Présenté par : Jean-Pierre Demailly

Qiʼan Guan ¹ ; Xiangyu Zhou ² ; Langfeng Zhu ¹

¹ Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
² Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China

Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 797-800.

Résumé
Abstract

Dans cette Note, nous améliorons lʼestimation des constantes dans la généralisation par Ohsawa du théorème dʼextension $L^{2}$ de Ohsawa–Takegoshi concernant les fonctions holomorphes, et nous appliquons ce résultat à lʼétude de la conjecture de Suita. Nous présentons également une remarque permettant de généraliser le théorème dʼextension de Ohsawa–Takegoshi au cas des $(n - 1, q)$ -formes lisses $\bar{\partial}$ -fermées. Enfin, nous montrons que le facteur tordu dans lʼidentité tordue de Bochner–Kodaira peut être une fonction plurisuperharmonique non lisse.

In this Note, we improve the estimate in Ohsawaʼs generalization of the Ohsawa–Takegoshi $L^{2}$ extension theorem for holomorphic functions by finding a smaller constant, and apply the result to the Suita conjecture. We also present a remark allowing to generalize the Ohsawa–Takegoshi extension theorem to the case of $\bar{\partial}$ -closed smooth $(n - 1, q)$ -forms. Finally, we prove that the twist factor in the twisted Bochner–Kodaira identity can be a non-smooth plurisuperharmonic function.

Reçu le : 2011-01-13
Accepté le : 2011-06-01
Publié le : 2011-06-22

DOI : 10.1016/j.crma.2011.06.001

Affiliations des auteurs :

Qiʼan Guan ¹ ; Xiangyu Zhou ² ; Langfeng Zhu ¹

¹ Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
² Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China

@article{CRMATH_2011__349_13-14_797_0,
     author = {Qi'an Guan and Xiangyu Zhou and Langfeng Zhu},
     title = {On the {Ohsawa{\textendash}Takegoshi} $ {L}^{2}$ extension theorem and the twisted {Bochner{\textendash}Kodaira} identity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {797--800},
     publisher = {Elsevier},
     volume = {349},
     number = {13-14},
     year = {2011},
     doi = {10.1016/j.crma.2011.06.001},
     language = {en},
}

TY  - JOUR
AU  - Qiʼan Guan
AU  - Xiangyu Zhou
AU  - Langfeng Zhu
TI  - On the Ohsawa–Takegoshi $ {L}^{2}$ extension theorem and the twisted Bochner–Kodaira identity
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 797
EP  - 800
VL  - 349
IS  - 13-14
PB  - Elsevier
DO  - 10.1016/j.crma.2011.06.001
LA  - en
ID  - CRMATH_2011__349_13-14_797_0
ER  -

%0 Journal Article
%A Qiʼan Guan
%A Xiangyu Zhou
%A Langfeng Zhu
%T On the Ohsawa–Takegoshi $ {L}^{2}$ extension theorem and the twisted Bochner–Kodaira identity
%J Comptes Rendus. Mathématique
%D 2011
%P 797-800
%V 349
%N 13-14
%I Elsevier
%R 10.1016/j.crma.2011.06.001
%G en
%F CRMATH_2011__349_13-14_797_0

Qiʼan Guan; Xiangyu Zhou; Langfeng Zhu. On the Ohsawa–Takegoshi $ {L}^{2}$ extension theorem and the twisted Bochner–Kodaira identity. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 797-800. doi : 10.1016/j.crma.2011.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.06.001/

Bibliographie
Cité par

[1] B. Berndtsson The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman, Ann. Inst. Fourier (Grenoble), Volume 46 (1996) no. 4, pp. 1083-1094

[2] Z. Blocki Some estimates for the Bergman kernel and metric in terms of logarithmic capacity, Nagoya Math. J., Volume 185 (2007), pp. 143-150

[3] B. Chen On the extension of $L^{2}$ holomorphic $(p, q)$ -forms, Chin. Ann. Math. Ser. A, Volume 19 (1998), pp. 433-436 (in Chinese)

[4] B. Chen A remark on an extension theorem of Ohsawa, Chin. Ann. Math. Ser. A, Volume 24 (2003), pp. 129-134 (in Chinese)

[5] J.-P. Demailly On the Ohsawa–Takegoshi–Manivel $L^{2}$ extension theorem, Paris, September 1997 (Progress in Mathematics) (2000)

[6] J.-P. Demailly Complex analytic and differential geometry http://www-fourier.ujf-grenoble.fr/~demailly/books.html (electronically accessible at)

[7] J.-P. Demailly Analytic Methods in Algebraic Geometry, Higher Education Press, Beijing, 2010

[8] H. Donnelly; C. Fefferman $L^{2}$ -cohomology and index theorem for the Bergman metric, Ann. of Math. (2), Volume 118 (1983), pp. 593-618

[9] H. Donnelly; F. Xavier On the differential form spectrum of negatively curved Riemann manifolds, Amer. J. Math., Volume 106 (1984), pp. 169-185

[10] S. Giret Prolongement de courants positifs à travers une sous-variété CR, J. Math. Pures Appl. (9), Volume 78 (1999) no. 9, pp. 895-911

[11] Q. Guan, X. Zhou, L. Zhu, On the Ohsawa–Takegoshi $L^{2}$ extension theorem and the twisted Bochner–Kodaira identity with a non-smooth twist factor, preprint, 2010.

[12] R. Harvey Removable singularities for positive currents, Amer. J. Math., Volume 96 (1974), pp. 67-78

[13] L. Manivel Un théorème de prolongement $L^{2}$ de sections holomorphes dʼun fibré vectoriel, Math. Z., Volume 212 (1993), pp. 107-122

[14] J. McNeal; D. Varolin Analytic inversion of adjunction: $L^{2}$ extension theorems with gain, Ann. Inst. Fourier (Grenoble), Volume 57 (2007) no. 3, pp. 703-718

[15] T. Ohsawa On the extension of $L^{2}$ holomorphic functions. III. Negligible weights, Math. Z., Volume 219 (1995) no. 2, pp. 215-225

[16] T. Ohsawa; K. Takegoshi On the extension of $L^{2}$ holomorphic functions, Math. Z., Volume 195 (1987), pp. 197-204

[17] Y.-T. Siu Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Differential Geom., Volume 17 (1982), pp. 55-138

[18] Y.-T. Siu The Fujita conjecture and the extension theorem of Ohsawa–Takegoshi, Geometric Complex Analysis, Hayama, World Scientific, 1996, pp. 577-592

[19] N. Suita Capacities and kernels on Riemann surfaces, Arch. Ration. Mech. Anal., Volume 46 (1972), pp. 212-217

Cité par Sources :

Commentaires - Politique