Comptes Rendus
Differential geometry
Bergman kernels on punctured Riemann surfaces
Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1018-1022.

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.

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.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2016.08.006
Hugues Auvray 1; Xiaonan Ma 2; George Marinescu 3

1 Laboratoire de mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, Département de mathématiques, bâtiment 425, 91405 Orsay, France
2 Université Paris-Diderot – Paris-7, UFR de mathématiques, case 7012, 75205 Paris cedex 13, France
3 Universität zu Köln, Mathematisches Institut, Weyertal 86–90, 50931 Köln, Germany
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Hugues Auvray; Xiaonan Ma; George Marinescu. Bergman kernels on punctured Riemann surfaces. Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1018-1022. doi : 10.1016/j.crma.2016.08.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.08.006/

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