Kähler families of Green’s functions

Vincent Guedj; Tat Dat Tô

doi:10.5802/jep.291

Kähler families of Green’s functions
[Familles de fonctions de Green kählériennes]

Vincent Guedj ¹ ; Tat Dat Tô ²

¹ Institut Universitaire de France et Institut de Mathématiques de Toulouse, Université de Toulouse, 118 route de Narbonne, 31400 Toulouse, France
² Sorbonne Université, Institut de mathématiques de Jussieu – Paris Rive Gauche, 4, place Jussieu, 75252 Paris Cedex 05, France.

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 319-339.

Résumé
Abstract

Dans une série remarquable de travaux, Guo, Phong, Song et Sturm ont obtenu d’importantes estimations uniformes pour les fonctions de Green associées à certaines métriques de Kähler. Dans cette note, nous élargissons le champ d’application de leurs techniques en supprimant l’une de leurs hypothèses et en permettant à la structure complexe de varier. Nous appliquons nos résultats à diverses familles de métriques kählériennes canoniques.

In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green’s functions associated with certain Kähler metrics. In this note, we broaden the scope of their techniques by removing one of their assumptions and allowing the complex structure to vary. We apply our results to various families of canonical Kähler metrics.

Reçu le : 2024-06-03
Accepté le : 2025-02-11
Publié le : 2025-02-18

DOI : 10.5802/jep.291

Classification : 32W20, 32U05, 32Q15, 35A23
Keywords: Green’s function, Monge-Ampère equation, a priori estimates
Mots-clés : Fonctions de Green, équation de Monge-Ampère, estimées a priori

Affiliations des auteurs :

Vincent Guedj ¹ ; Tat Dat Tô ²

¹ Institut Universitaire de France et Institut de Mathématiques de Toulouse, Université de Toulouse, 118 route de Narbonne, 31400 Toulouse, France
² Sorbonne Université, Institut de mathématiques de Jussieu – Paris Rive Gauche, 4, place Jussieu, 75252 Paris Cedex 05, France.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Vincent Guedj and Tat Dat T\^o},
     title = {K\"ahler families of {Green{\textquoteright}s} functions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {319--339},
     publisher = {\'Ecole polytechnique},
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     year = {2025},
     doi = {10.5802/jep.291},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.291/}
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Vincent Guedj; Tat Dat Tô. Kähler families of Green’s functions. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 319-339. doi : 10.5802/jep.291. https://jep.centre-mersenne.org/articles/10.5802/jep.291/

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