A Yau–Tian–Donaldson correspondence on a class of toric fibrations

Jubert, Simon

doi:10.5802/aif.3580

A Yau–Tian–Donaldson correspondence on a class of toric fibrations
[Une correspondance du type Yau–Tian–Donaldson sur une large classe de fibrés de toriques]

Jubert, Simon ^1,²

¹ Departement de Mathématiques UQAM, C.P. 8888 Succursale Centre-ville, Montréal (Quebec) H3C 3P8 (Canada)
² Institut de Mathématiques de Toulouse Université Paul Sabatier 118 route de Narbonne 31062, Toulouse (France)

Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2567-2604.

Résumé
Abstract

Nous établissons une correspondance du type Yau–Tian–Donaldson, exprimée en terme d’un polytope de Delzant, concernant l’existence de métriques Kähler extrémales sur une large classe de fibrations toriques définie par Apostolov–Calderbank–Gauduchon–Tonnesen-Friedman et appelée semi-simple principal toric fibrations. Nous utilisons qu’une extrémale sur l’espace total correspond à une métrique à courbure scalaire constante pondérée (dans le sens de Lahdili) sur la fibre torique correspondante pour obtenir une équivalence entre l’existence des métriques extrémales sur l’espace total et une notion appropriée de K-stabilité uniforme pondéree du polytope de Delzant correspondant. En tant qu’application, nous montrons que le fibré en plan projectif $ℙ (ℒ_{0} \oplus ℒ_{1} \oplus ℒ_{2})$ , où les $ℒ_{i}$ sont des fibrés holomorphes au dessus d’une courbe elliptique, admet une métrique extrémale dans chaque classe de Kähler.

We establish a Yau–Tian–Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal Kähler metrics on a large class of toric fibrations, introduced by Apostolov–Calderbank–Gauduchon–Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature Kähler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal Kähler metrics on the total space and a suitable notion of weighted uniform K-stability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle $ℙ (ℒ_{0} \oplus ℒ_{1} \oplus ℒ_{2})$ , where $ℒ_{i}$ are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.

Reçu le : 2021-08-27
Révisé le : 2022-02-15
Accepté le : 2022-05-12
Publié le : 2023-10-26

DOI : 10.5802/aif.3580

Classification : 00X99
Keywords: Differential geometry, complex geometry, complex analysis, toric geometry.
Mot clés : Géométrie différentielle, géométrie complexe, fibration toric principal semisimple, métrique Kähler extrémale, métrique à csc pondérée, K-stabilité uniforme.

Affiliations des auteurs :

Jubert, Simon ^{1, 2}

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Jubert, Simon. A Yau–Tian–Donaldson correspondence on a class of toric fibrations. Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2567-2604. doi : 10.5802/aif.3580. https://aif.centre-mersenne.org/articles/10.5802/aif.3580/

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