2025/03/03 by Thibaut Delcroix, Delcroix, Thibaut, Simon Jubert +1
Mathematics · Physics and Astronomy · #32Q20 #53C55 #Advanced Differential Geometry Research #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2503.01680
openalex publication_date 2025/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In K-stability, the delta invariant of a Fano variety encodes the existence of Kähler-Einstein metrics. We introduce a weighted analytic delta invariant, and a reduced version, that characterize the existence of weighted solitons. We further prove a sufficient condition of existence of weighted cscK metrics in terms of this invariant. We elucidate the relation between the weighted delta invariant and the greatest lower bound on the weighted Ricci curvature, called the weighted beta invariant. We provide a general upper bound for the weighted beta invariant in terms of moment images. Finally, we investigate how the geometry of semisimple principal fibrations, whose basis is not assumed to be cscK, allows to estimate their beta invariant in terms of the basis and the weighted fiber. Most of our statements are new even in the trivial weights settings, that is, for Kähler-Einstein and cscK metrics.