Numerical invariants for weighted cscK metrics
Thibaut Delcroix, Simon Jubert
TL;DR
The paper develops a comprehensive weighted Kähler framework by introducing a weighted analytic delta invariant δ_v and its reduced version to characterize weighted solitons and the existence of weighted cscK metrics. It connects δ_v to the greatest weighted Ricci lower bound β_v and provides a general upper bound via moment polytope data, while establishing a Chen–Tian-type decomposition that links coercivity to weighted Mabuchi functional positivity and the vanishing of the weighted Futaki invariant. Existence of weighted cscK metrics is proved through a weighted J-equation approach, including a robust set of a priori estimates and a weighted continuity method. Finally, the authors analyze semisimple principal fibrations, giving explicit formulas for the compatible beta invariants that relate the total space to the fiber and base geometry, with detailed results for P^1-bundles and comparisons to Zhang–Zhou’s work, highlighting the role of fiber geometry in governing global canonical metrics.
Abstract
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.