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We study the weighted constant scalar curvature, a modified scalar curvatureintroduced by Lahdili depending on weight functions $(v, w)$, on certainnon-compact semisimple toric fibrations, a generalization of the Calabi Ansatzdefined by Apostolov--Calderbank--Gauduchon--T{\o}nnesen-Friedman. We show thatthe natural analog of the weighted Futaki invariant of Lahdili can underreasonable assumptions be interpreted on an unbounded polyhedron $P \subset\mathbb{R}^n$ associated to $M$. In particular, we fix a certain class$\mathcal{W}$ of weights $(v, w)$, and prove that if $M$ admits a weighted cscKmetric, then $P$ is K-stable, and we give examples of weights on $\mathbb{C}^2$for which the weighted Futaki invariant vanishes but do not admit $(v, w)$-cscKmetrics. Following Jubert, we introduce a weighted Mabuchi energy$\mathcal{M}_{v,w}$ and show that the existence of a $(v, w)$-cscK metricimplies that it $\mathcal{M}_{v,w}$ proper, and prove a uniqueness result usingthe method of Guan. We show that weighted K-stability of the abstract fiber$\mathbb{C}$ is sufficient for the existence of weighted cscK metrics on thetotal space of line bundles $L \rightarrow B$ over a compact K\"ahler base,extending a result of Lahdili in the $\mathbb{P}^1$-bundles case. The rightchoice of weights corresponds to the (shrinking) K\"ahler-Ricci solitonequation, and we give an interpretation of the asyptotic geometry in this case.

Weighted K-stability for a class of non-compact toric fibrations