We study the weighted constant scalar curvature, a modified scalar curvature introduced by Lahdili (Proc Lond Math Soc 119(4):1065–1114, 2019) depending on weight functions (v,w)\documentclass[12pt]{minimal}
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\begin{document}$$(v, \, w)$$\end{document}, on non-compact semisimple principal toric fibrations. The latter notion is a generalization of the Calabi Ansatz originally defined by Apostolov et al. (J Differ Geom 68(2):277–345, 2004). This setup turns out to reduce the weighted cscK problem on the total space to a different weighted cscK problem on a fixed toric fiber M. We show that the natural analog of the weighted Futaki invariant of Lahdili (Proc Lond Math Soc 119(4):1065–1114, 2019) can under reasonable assumptions be interpreted on an unbounded polyhedron P⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$P \subset {\mathbb {R}}^n$$\end{document} associated to M. In particular, we fix a certain class W\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {W}}$$\end{document} of weights (v,w)\documentclass[12pt]{minimal}
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\begin{document}$$(v, \,w)$$\end{document} and prove that if M admits a weighted cscK metric, then P is K-stable, and we give examples of weights on C2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^2$$\end{document} for which the weighted Futaki invariant vanishes but do not admit (v,w)\documentclass[12pt]{minimal}
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\begin{document}$$(v,\, w)$$\end{document}-cscK metrics. Following Jubert (A Yau-Tian-Donaldson correspondence on a class of toric fibrations. arXiv:2108.12297, 2021), we introduce a weighted Mabuchi energy Mv,w\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}_{v,w}$$\end{document} and show that the existence of a (v,w)\documentclass[12pt]{minimal}
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\begin{document}$$(v, \, w)$$\end{document}-cscK metric implies that it Mv,w\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}_{v,w}$$\end{document} proper. The well-definedness of Mv,w\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}_{v,w}$$\end{document} in this setting also allows us to prove a uniqueness result using the method of Guan (Math Res Lett 6:547–555, 1999). As an application, we show that weighted K-stability of the abstract fiber C\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}$$\end{document} is sufficient for the existence of weighted cscK metrics on the total space of line bundles L→B\documentclass[12pt]{minimal}
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\begin{document}$$L \rightarrow B$$\end{document} over a compact Kähler base, extending the result in Lahdili (Proc Lond Math Soc 119(4):1065–1114, 2019) in the P1\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {P}}^1$$\end{document}-bundles case. As a consequence, we recover a well-known existence result for shrinking Kähler–Ricci solitons (Feldman et al. in J Differ Geom 65(2):169–209, 2003; Futaki and Wang in Asian J Math 15:33–52, 2011; Li in On rotationally symmetric Kähler-Ricci solitons. arXiv:1004.4049, 2011). Finally, we give some interpretations in terms of asymptotic geometry.
