Generalized $ {L}^{2}$ extension theorem and a conjecture of Ohsawa

Guan, Qiʼan; Zhou, Xiangyu

doi:10.1016/j.crma.2013.01.012

Analytic Geometry

Generalized

L^{2}

extension theorem and a conjecture of Ohsawa

Présenté par : Jean-Pierre Demailly

Guan, Qiʼan ¹ ; Zhou, Xiangyu ²

¹ Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China
² Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China

Comptes Rendus. Mathématique, Tome 351 (2013) no. 3-4, pp. 111-114.

Résumé
Abstract

Dans cet article, nous déterminons la constante optimale intervenant dans lʼestimée du théorème dʼextension $L^{2}$ généralisé de Ohsawa. Le résultat vaut pour les fibrés vectoriels holomorphes sur une classe de variétés complexes incluant à la fois les variétés de Stein et les variétés algébriques projectives complexes. Comme application, nous obtenons la solution dʼune conjecture correspondante dʼOhsawa.

In this paper, we determine the optimal constant in the estimate of Ohsawaʼs generalized $L^{2}$ extension theorem. The result holds for holomorphic vector bundles on a class of complex manifolds including both Stein manifolds and complex projective algebraic manifolds. As an application, we obtain a solution to a related conjecture of Ohsawa.

Reçu le : 2012-11-30
Accepté le : 2013-01-30
Publié le : 2013-02-01

DOI : 10.1016/j.crma.2013.01.012

Affiliations des auteurs :

Guan, Qiʼan ¹ ; Zhou, Xiangyu ²

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Guan, Qiʼan; Zhou, Xiangyu. Generalized $ {L}^{2}$ extension theorem and a conjecture of Ohsawa. Comptes Rendus. Mathématique, Tome 351 (2013) no. 3-4, pp. 111-114. doi : 10.1016/j.crma.2013.01.012. http://www.numdam.org/articles/10.1016/j.crma.2013.01.012/

Bibliographie
Cité par

[1] Berndtsson, B. 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, On the Ohsawa–Takegoshi extension theorem, preprint, 2012.

[3] Z. Blocki, Suita conjecture and the Ohsawa–Takegoshi extension theorem, Inventiones Mathematicae, , in press. | DOI

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

[5] Demailly, J.-P. Estimations $L^{2}$ pour lʼopérateur dʼun fibré vectoriel holomorphe semi-positif au-dessus dʼune variété kählérienne complète, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982) no. 3, pp. 457-511 (in French)

[6] Demailly, J.-P. Fonction de Green pluricomplexe et mesures pluriharmoniques, Séminaire de théorie spectrale et géométrie, vol. 4, Univ. Grenoble-1, Saint-Martin-dʼHères, 1986, pp. 131-143 (in French), année 1985–1986

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

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

[9] Demailly, J.P. Kähler manifolds and transcendental techniquesin algebraic geometry, International Congress of Mathematicians 2006, vol. I, Eur. Math. Soc., Zürich, 2007, pp. 153-186

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

[11] Fornæss, J.E.; Narasimhan, R. The Levi problem on complex spaces with singularities, Math. Ann., Volume 248 (1980) no. 1, pp. 47-72

[12] Guan, Q.A.; Zhou, X.Y. Optimal constant problem in the $L^{2}$ extension theorem, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) no. 15–16, pp. 753-756

[13] Q.A. Guan, X.Y. Zhou, Optimal constant in $L^{2}$ extension and a proof of a conjecture of Ohsawa, preprint.

[14] Guan, Q.A.; Zhou, X.Y.; Zhu, L.F. On the Ohsawa–Takegoshi $L^{2}$ extension theorem and the twisted Bochner–Kodaira identity, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 13–14, pp. 797-800

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

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

[17] Ohsawa, T. On the extension of $L^{2}$ holomorphic functions. II, Publ. Res. Inst. Math. Sci., Volume 24 (1988) no. 2, pp. 265-275

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

[19] Ohsawa, T. On the extension of $L^{2}$ holomorphic functions. IV. A new density concept, Geometry and Analysis on Complex Manifolds, World Sci. Publ., River Edge, NJ, 1994, pp. 157-170

[20] Ohsawa, T.; Ohsawa, T. Erratum to: “On the extension of $L^{2}$ holomorphic functions. V. Effects of generalization” [Nagoya Math. J. 161 (2001) 1–21], Nagoya Math. J., Volume 161 (2001), pp. 1-21

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

[22] Sario, L.; Oikawa, K. Capacity functions, Die Grundlehren der mathematischen Wissenschaften, vol. 149, Springer-Verlag Inc., New York, 1969

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

[24] Siu, Y.-T. Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Göttingen, 2000, Springer, Berlin (2002), pp. 223-277

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

[26] Zhu, L.F.; Guan, Q.A.; Zhou, X.Y. On the Ohsawa–Takegoshi $L^{2}$ extension theorem and the Bochner–Kodaira identity with non-smooth twist factor, J. Math. Pures Appl. (9), Volume 97 (2012) no. 6, pp. 579-601

Cité par Sources :

^☆ The second author was partially supported by NSFC.