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 [35] depending on weight functions (v, w), on certain non-compact semisimple toric fibrations. The latter notion is a generalization of the Calabi Ansatz originally defined by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman [2]. The 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 [35] can under reasonable assumptions be interpreted on an unbounded polyhedron P ⊂ R n associated to M . In particular, we fix a certain class 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 C 2 for which the weighted Futaki invariant vanishes but do not admit (v, w)-cscK metrics. Following [34], we introduce a weighted Mabuchi energy Mv,w and show that the existence of a (v, w)-cscK metric implies that it Mv,w proper. The well-definedness of Mv,w in this setting also allows us to prove a uniqueness result using the method of [29]. We apply the theory in a few special cases and make connections with asymptotic geometry. In particular, we show that weighted K-stability of the abstract fiber C is sufficient for the existence of weighted cscK metrics on the total space of line bundles L → B over a compact Kähler base, extending the result in [35] in the 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.