[Noyau de Bergman sur une surface de Riemann épointée]
On considère une surface de Riemann compacte munie d'une métrique hermitienne singulière égale à la métrique de Poincaré du disque épointé près d'un diviseur donné. On considère un fibré en droites holomorphe avec une métrique singulière qui polarise la métrique (singulière) sur la surface de Riemann. On donne l'asymptotique explicite près des singularités de la fonction de Bergman lorsque la puissance de fibré en droites tend vers l'infini.
In this paper, we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincaré metric near the punctures and a holomorphic line bundle that polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincaré metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel function, involving a fractional growth order of the tensor power.
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@article{CRMATH_2016__354_10_1018_0,
author = {Auvray, Hugues and Ma, Xiaonan and Marinescu, George},
title = {Bergman kernels on punctured {Riemann} surfaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {1018--1022},
publisher = {Elsevier},
volume = {354},
number = {10},
year = {2016},
doi = {10.1016/j.crma.2016.08.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.08.006/}
}
TY - JOUR AU - Auvray, Hugues AU - Ma, Xiaonan AU - Marinescu, George TI - Bergman kernels on punctured Riemann surfaces JO - Comptes Rendus. Mathématique PY - 2016 SP - 1018 EP - 1022 VL - 354 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.08.006/ DO - 10.1016/j.crma.2016.08.006 LA - en ID - CRMATH_2016__354_10_1018_0 ER -
%0 Journal Article %A Auvray, Hugues %A Ma, Xiaonan %A Marinescu, George %T Bergman kernels on punctured Riemann surfaces %J Comptes Rendus. Mathématique %D 2016 %P 1018-1022 %V 354 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.08.006/ %R 10.1016/j.crma.2016.08.006 %G en %F CRMATH_2016__354_10_1018_0
Auvray, Hugues; Ma, Xiaonan; Marinescu, George. Bergman kernels on punctured Riemann surfaces. Comptes Rendus. Mathématique, Tome 354 (2016) no. 10, pp. 1018-1022. doi : 10.1016/j.crma.2016.08.006. http://www.numdam.org/articles/10.1016/j.crma.2016.08.006/
[1] Comparaison des métriques d'Arakelov et de Poincaré sur , Duke Math. J., Volume 80 (1995), pp. 295-307
[2] The space of Poincaré type Kähler metrics, J. Reine Angew. Math. (2016) (in press) | DOI
[3] Bergman kernels on punctured Riemann surfaces (40 p.) | arXiv
[4] Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse), Ann. Sci. Éc. Norm. Supér. (4), Volume 30 (1997), pp. 41-96
[5] Complex immersions and Quillen metrics, Publ. Math. IHÉS, Volume 74 (1991) ii+298 p. (1992)
[6] On the asymptotic expansion of Bergman kernel, J. Differ. Geom., Volume 72 (2006), pp. 1-41
[7] Uniform sup-norm bounds on average for cusp forms of higher weights, 2013 (preprint) | arXiv
[8] Bounding the sup-norm of automorphic forms, Geom. Funct. Anal., Volume 14 (2004), pp. 1267-1277
[9] Holomorphic Morse Inequalities and Bergman Kernels, Prog. Math., vol. 254, Birkhäuser Verlag, Basel, Switzerland, 2007
[10] Points de petite hauteur sur les courbes modulaires , Invent. Math., Volume 131 (1998), pp. 645-674
[11] Hirzebruch's proportionality theorem in the noncompact case, Invent. Math., Volume 42 (1977), pp. 239-272
[12] On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., Volume 32 (1990), pp. 99-130
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