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no. 5
An explicit estimate of the Bergman kernel for positive line bundles
Xu Wang1
1 Departement of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 805-816.

We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tian’s partial C 0 -estimate.

Nous donnerons une estimation explicite de la borne inférieure du noyau de Bergman associé à un fibré de droites positif. Dans le cas de la surface compacte de Riemann, notre résultat peut être vu comme une version explicite de l’estimation partielle C 0 de Tian.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1752
Classification: 32A25
Keywords: Bergman kernel, Ohsawa–Takegoshi theorem
Mot clés : Noyau Bergman, Théorème d’Ohsawa–Takegoshi

Xu Wang 1

1 Departement of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Xu Wang. An explicit estimate of the Bergman kernel  for positive line bundles. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 805-816. doi : 10.5802/afst.1752. https://afst.centre-mersenne.org/articles/10.5802/afst.1752/

[1] Richard H. Bamler Convergence of Ricci flows with bounded scalar curvature, Ann. Math., Volume 188 (2018) no. 3, pp. 753-831 | MR | Zbl

[2] Bo Berndtsson Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Volume 169 (2009) no. 2, pp. 531-560 | DOI | MR | Zbl

[3] 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 | DOI | MR

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

[5] Zbigniew Błocki Cauchy–Riemann meet Monge-Ampère, Bull. Math. Sci., Volume 4 (2014) no. 3, pp. 433-480 | DOI

[6] Zbigniew Błocki; Włodzimierz Zwonek Estimates for the Bergman Kernel and the Multidimensional Suita Conjecture (2014) | arXiv

[7] Xiuxiong Chen; Bing Wang Space of Ricci flows I, Pure Appl. Math., Volume 65 (2012) no. 10, pp. 1399-1457 | DOI | MR

[8] Xiuxiong Chen; Bing Wang Space of Ricci flows II (2014) | arXiv

[9] Jean-Pierre Demailly Singular hermitian metrics on positive line bundles, Complex algebraic varieties (Lecture Notes in Mathematics), Volume 1507, Springer, 1992, pp. 87-104 | MR | Zbl

[10] Simon Donaldson; Song Sun Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, Algebr. Geom., Volume 213 (2014) no. 1, pp. 63-106

[11] Robert E. Greene; Hung-Hsi Wu Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, 699, Springer, 1979 | DOI

[12] 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

[13] Ernst Heintze; Hermann Karcher A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Éc. Norm. Supér., Volume 11 (1978) no. 4, pp. 451-470 | DOI | Numdam | MR

[14] Wenshuai Jiang Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture, J. Reine Angew. Math., Volume 717 (2016), pp. 195-226 | DOI | MR

[15] Wenshuai Jiang; Feng Wang; Xiaohua Zhu Bergman kernels for a sequence of almost Kähler-Ricci solitons, Ann. Inst. Fourier, Volume 67 (2017) no. 3, pp. 1279-1320 | DOI | Numdam

[16] Gang Liu; Gábor Székelyhidi Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below (2018) | arXiv

[17] Frank Morgan; David L. Johnson Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J., Volume 49 (2000) no. 3, pp. 1017-1041 | MR | Zbl

[18] T. Nguyen; X. Wang On and around the Berndtsson-Lempert method in Ohsawa-Takegoshi theory (to appear)

[19] Takeo Ohsawa; Kensho Takegoshi On the extension of L 2 -holomorphic functions, Math. Z., Volume 195 (1987), pp. 197-204 | DOI | Zbl

[20] Gábor Székelyhidi The partial C 0 -estimate along the continuity method, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 537-560 | DOI | MR

[21] Gang Tian On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., Volume 32 (1990) no. 1, pp. 99-130

[22] Gang Tian On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., Volume 101 (1990) no. 1, pp. 101-172 | DOI | MR

[23] Gang Tian Partial C 0 -estimates for Kähler-Einstein metrics, Commun. Math. Stat., Volume 1 (2013) no. 2, pp. 105-113 | DOI | Zbl

[24] Feng Wang; Xiaohua Zhu Tian’s partial C 0 -estimate implies Hamilton–Tian’s conjecture (2020) | arXiv

[25] Z. Wang The sphere theorem (available in http://staff.ustc.edu.cn/~wangzuoq/Courses/16S-RiemGeom/Notes/Lec22.pdf)

[26] Kewei Zhang Some refinements of the partial C 0 -estimate (2019) | arXiv

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