An explicit estimate of the Bergman kernel for positive line bundles

Wang, Xu

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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.

Classification : 32A25
Keywords: Bergman kernel, Ohsawa–Takegoshi theorem
Mots-clés : Noyau Bergman, Théorème d’Ohsawa–Takegoshi

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

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

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

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

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

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

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

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

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

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

[12] Guan, Qi’an; Zhou, Xiangyu 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] Heintze, Ernst; Karcher, Hermann 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 | MR | Numdam

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

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

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

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

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

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

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

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

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

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

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

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

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