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 , where are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.
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 , où les sont des fibrés holomorphes au dessus d’une courbe elliptique, admet une métrique extrémale dans chaque classe de Kähler.
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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.
Jubert, Simon 1, 2
@article{AIF_2023__73_6_2567_0, author = {Jubert, Simon}, title = {A {Yau{\textendash}Tian{\textendash}Donaldson} correspondence on a class of toric fibrations}, journal = {Annales de l'Institut Fourier}, pages = {2567--2604}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {6}, year = {2023}, doi = {10.5802/aif.3580}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3580/} }
TY - JOUR AU - Jubert, Simon TI - A Yau–Tian–Donaldson correspondence on a class of toric fibrations JO - Annales de l'Institut Fourier PY - 2023 SP - 2567 EP - 2604 VL - 73 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3580/ DO - 10.5802/aif.3580 LA - en ID - AIF_2023__73_6_2567_0 ER -
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Jubert, Simon. A Yau–Tian–Donaldson correspondence on a class of toric fibrations. Annales de l'Institut Fourier, Volume 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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