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

Abstract

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--Tønnesen-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 $L\to B$ over a compact Kähler base, extending a result of Lahdili in the ${\mathbb{P}}^1$-bundles case. The right choice of weights corresponds to the (shrinking) Kähler-Ricci soliton equation, and we give an interpretation of the asyptotic geometry in this case.

Added Corollary 1.3 and its proof, and other minor changes, as per the suggestion of the referee. Final version, to appear in J. Geom. Anal