Complex analysis
${L}^{2}$ extension theorem for jets with variable denominators
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 181-193.

By studying the variable denominators introduced by X. Zhou–L. Zhu, we generalize the results of D. Popovici for the ${L}^{2}$ extension theorem for jets. As a direct corollary, we also give a generalization of T. Ohsawa–K. Takegoshi’s extension theorem to a jet version.

Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.167
Classification: 32D15,  32L10,  32Q15,  32T35
Keywords: Continuation of analytic objects in several complex variables; Sheaves and cohomology of sections of holomorphic vector bundles, general results, Kähler manifolds, Exhaustion functions
Sheng Rao 1, 2; Runze Zhang 3

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
2. Université de Grenoble-Alpes, Institut Fourier (Mathématiques) UMR 5582 du C.N.R.S., 100 rue des Maths, 38610 Gières, France
3. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China.
@article{CRMATH_2021__359_2_181_0,
author = {Sheng Rao and Runze Zhang},
title = {$L^2$ extension theorem for jets with variable denominators},
journal = {Comptes Rendus. Math\'ematique},
pages = {181--193},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {2},
year = {2021},
doi = {10.5802/crmath.167},
language = {en},
}
TY  - JOUR
AU  - Sheng Rao
AU  - Runze Zhang
TI  - $L^2$ extension theorem for jets with variable denominators
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 181
EP  - 193
VL  - 359
IS  - 2
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.167
DO  - 10.5802/crmath.167
LA  - en
ID  - CRMATH_2021__359_2_181_0
ER  - 
%0 Journal Article
%A Sheng Rao
%A Runze Zhang
%T $L^2$ extension theorem for jets with variable denominators
%J Comptes Rendus. Mathématique
%D 2021
%P 181-193
%V 359
%N 2
%U https://doi.org/10.5802/crmath.167
%R 10.5802/crmath.167
%G en
%F CRMATH_2021__359_2_181_0
Sheng Rao; Runze Zhang. $L^2$ extension theorem for jets with variable denominators. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 181-193. doi : 10.5802/crmath.167. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.167/

[1] Bo Berndtsson; László Lempert A proof of the Ohsawa–Takegoshi theorem with sharp estimates, J. Math. Soc. Japan, Volume 68 (2016) no. 4, pp. 1461-1472 | Article | MR 3564439 | Zbl 1360.32006

[2] Zbigniew Błocki Suita conjecture and the Ohsawa–Takegoshi extension theorem, Invent. Math., Volume 193 (2013) no. 1, pp. 149-158 | Article | MR 3069114 | Zbl 1282.32014

[3] Jian Chen; Sheng Rao ${L}^{2}$ extension of $\overline{\partial }$-closed forms on weakly pseudoconvex Kähler manifolds (2020) (https://arxiv.org/abs/2009.14101)

[4] Jean-Pierre Demailly Estimations ${L}^{2}$ pour l’opérateur $\overline{\partial }$ d’un fibré vectoriel holomorphe semi-positif au dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982), pp. 457-511 | Article | Numdam | Zbl 0507.32021

[5] Jean-Pierre Demailly On the Ohsawa–Takegoshi–Manivel ${L}^{2}$ extension theorem, Complex analysis and geometry (Paris, 1997) (Progress in Mathematics), Volume 188, Birkhäuser, 2000, pp. 47-82 | Article | MR 1782659 | Zbl 0959.32019

[6] Jean-Pierre Demailly Extension of holomorphic functions defined on non reduced analytic subvarieties, The legacy of Bernhard Riemann after one hundred and fifty years. Vol. I (Advanced Lectures in Mathematics (ALM)), Volume 35, International Press, 2016, pp. 191-222 | MR 3525916 | Zbl 1360.14025

[7] Qi An Guan; Xian Gyu 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 | Article | MR 3296822 | Zbl 1348.32008

[8] Qi An Guan; Xian Gyu Zhou; Lang Feng Zhu On the Ohsawa–Takegoshi ${L}^{2}$ extension theorem and the twisted Bochner–Kodaira identity, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 13-14, pp. 797-800 | Article | MR 2825944 | Zbl 1227.32014

[9] Genki Hosono The optimal jet ${L}^{2}$ extension of Ohsawa–Takegoshi type, Nagoya Math. J., Volume 239 (2020), pp. 153-172 | Article | MR 4138899 | Zbl 1452.32015

[10] Laurent Manivel Un théorème de prolongement ${L}^{2}$ de sections holomorphes d’un fibré hermitien, Math. Z., Volume 212 (1993) no. 1, pp. 107-122 | Article | Zbl 0789.32015

[11] Jeffery D. McNeal; Dror Varolin Analytic inversion of adjunction: ${L}^{2}$ extension theorems with gain, Ann. Inst. Fourier, Volume 57 (2007) no. 3, pp. 703-718 | Article | Numdam | MR 2336826 | Zbl 1208.32011

[12] Jeffery D. McNeal; Dror Varolin Extension of jets with ${L}^{2}$ estimates, and an application (2017) (https://arxiv.org/abs/1707.04483)

[13] Takeo Ohsawa; Kensho Takegoshi On the extension of ${L}^{2}$ holomorphic functions, Math. Z., Volume 195 (1987) no. 2, pp. 197-204 | Article | Zbl 0625.32011

[14] Dan Popovici ${L}^{2}$ extension for jets of holomorphic sections of a hermitian line bundle, Nagoya Math. J., Volume 180 (2005), pp. 1-34 | Article | MR 2186666 | Zbl 1116.32017

[15] Xian Gyu Zhou; Lang Feng Zhu An optimal ${L}^{2}$ extension theorem on weakly pseudoconvex Kähler manifolds, J. Differ. Geom., Volume 110 (2018) no. 1, pp. 135-186 | Article

[16] Xian Gyu Zhou; Lang Feng Zhu Siu’s lemma, optimal ${L}^{2}$ extension and applications to twisted pluricanonical sheaves, Math. Ann., Volume 377 (2020) no. 1-2, pp. 675-722 | Article | MR 4099619 | Zbl 1452.32014

[17] Lang Feng Zhu; Qi An Guan; Xian Gyu Zhou On the Ohsawa–Takegoshi ${L}^{2}$ extension theorem and the Bochner–Kodaira identity with non-smooth twist factor, J. Math. Pures Appl., Volume 97 (2012) no. 6, pp. 579-601 | MR 2921602 | Zbl 1244.32005

Cited by Sources: