We show that a compact weighted extremal Kähler manifold (as defined by the third named author in [50]) has coercive weighted Mabuchi energy with respect to a maximal complex torus G in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kähler metrics satisfying special curvature conditions, including Kähler metrics with constant scalar curvature [14,21], extremal Kähler metrics [44], Kähler-Ricci solitons [27] and their weighted extensions [12,42]. Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any non-product G-equivariant smooth Kähler test configuration with reduced central fibre, a property also known as weighted K-polystability on such test configurations. For a class of fibre-bundles, we use our result in conjunction with the results of Chen-Cheng [21], He [44] and in order to characterize the existence of extremal Kähler metrics and Calabi-Yau cones associated to the total space, in terms of the coercivity of the weighted Mabuchi energy of the fibre. In particular, this yields an existence result for Sasaki-Einstein metrics on Fano toric fibrations, extending the results of Futaki-Ono-Wang [36] in the toric Fano case, and of in the case of Fano P 1 -bundles.

We established 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(L0 ⊕ L1 ⊕ L2), where Li are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.

The second author has shown that existence of extremal Kähler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to prove various sufficient conditions of weighted uniform K-stability which can be checked effectively and explore the low dimensional new examples of extremal Kähler metrics it provides.

The second author has shown that existence of extremal Kähler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to prove various sufficient conditions of weighted uniform K-stability which can be checked effectively and explore the low dimensional new examples of extremal Kähler metrics it provides.

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 ⊕ L 1 ⊕ L 2 ), where L i are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.Résumé. -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 ⊕ L 1 ⊕ 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.