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 (Proc Lond Math Soc 119(4):1065–1114, 2019) depending on weight functions \((v, \, w)\) $(v,w)$ , 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 \(P \subset {\mathbb {R}}^n\) $P\subset {\mathbb{R}}^n$ associated to M. In particular, we fix a certain class \({\mathcal {W}}\) $\mathcal{W}$ of weights \((v, \,w)\) $(v,w)$ and prove that if M admits a weighted cscK metric, then P is K-stable, and we give examples of weights on \({\mathbb {C}}^2\) ${\mathbb{C}}^2$ for which the weighted Futaki invariant vanishes but do not admit \((v,\, w)\) $(v,w)$ -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 \({\mathcal {M}}_{v,w}\) ${\mathcal{M}}_{v,w}$ and show that the existence of a \((v, \, w)\) $(v,w)$ -cscK metric implies that it \({\mathcal {M}}_{v,w}\) ${\mathcal{M}}_{v,w}$ proper. The well-definedness of \({\mathcal {M}}_{v,w}\) ${\mathcal{M}}_{v,w}$ 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 \({\mathbb {C}}\) $\mathbb{C}$ is sufficient for the existence of weighted cscK metrics on the total space of line bundles \(L \rightarrow B\) $L\to B$ over a compact Kähler base, extending the result in Lahdili (Proc Lond Math Soc 119(4):1065–1114, 2019) in the \({\mathbb {P}}^1\) ${\mathbb{P}}^1$ -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.