2023/03/06 by Charles Cifarelli, Cifarelli, Charles · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2303.03263
openalex publication_date 2023/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ⊂ ℝn 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 ℂ2 for which the weighted Futaki invariant vanishes but do not admit (v, w)-cscK metrics. Following Jubert, 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, and prove a uniqueness result using the method of Guan. 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 a result of Lahdili in the ℙ1-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.