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

We study the weighted constant scalar curvature, a modified scalar curvature introduced by Lahdili depending on weight functions $(v, w)$, on certain non-compact semisimple toric fibrations, a generalization of the Calabi Ansatz defined by Apostolov--Calderbank--Gauduchon--T{\o}nnesen-Friedman. We show that the natural analog of the weighted Futaki invariant of Lahdili can under reasonable 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 cscK metric, 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 $(v, w)$-cscK metrics. Following Jubert, we introduce a weighted Mabuchi energy $\mathcal{M}_{v,w}$ and show that the existence of a $(v, w)$-cscK metric implies that it $\mathcal{M}_{v,w}$ proper, and prove a uniqueness result using the 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 the total 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 right choice of weights corresponds to the (shrinking) K\"ahler-Ricci soliton equation, and we give an interpretation of the asyptotic geometry in this case.