Let X be a compact Kähler manifold. In this paper we study the existence of constant weighted scalar curvature Kähler (weighted cscK) metrics on X. More precisely, we establish a priori C^k-estimates (k≥ 0) for the Kähler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.

We established a Yau--Tian--Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal K\"ahler 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 K\"ahler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal K\"ahler 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 $\mathbb{P}(\mathcal{L}_0\oplus\mathcal{L}_1 \oplus \mathcal{L}_2)$, where $\mathcal{L}_i$ are holomorphic line bundles over an elliptic curve, admits an extremal metric in every K\"ahler 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.

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature conditions, including constant scalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitons and their weighted extensions. Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any non-product equivariant smooth K\"ahler 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 recent results of Chen-Cheng, He, and Han-Li in order to characterize the existence of extremal Kahler 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 in the toric Fano case, and of Mabuchi-Nakagawa in the case of Fano projective line bundles.