Communications in Analysis and Geometry

Volume 30 (2022)

Number 4

Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles

Pages: 689 – 744



Vestislav Apostolov (Département de Mathématiques, Université du Québec à Montréal, Quebec, Canada; and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria)

Gideon Maschler (Department of Mathematics, Clark University, Worcester, Massachusetts, U.S.A.)

Christina W. Tønnesen-Friedman (Department of Mathematics, Union College, Schenectady, New York, U.S.A.)


We study the existence of weighted extremal Kähler metrics in the sense of [4, 32] on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined in [5], and they include the projective line bundles [29] and their blow-downs [31], thus providing a most general setting for extending the existence theory for extremal Kähler metrics pioneered by a seminal construction of Calabi [12]. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally Kähler, Einstein–Maxwell metrics of complex dimension $m \gt 2$, thus extending the recent constructions of [30, 38] to higher dimensions. For each admissible Kähler class on an admissible projective bundle, we associate an explicit function of one variable and show that if it is positive on the interval $(1,1)$, then there exists a weighted extremal Kähler metric in the given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is no Kähler metrics of constant weighted scalar curvature in that class. We also relate the positivity of the function to a notion of weighted K-stability, thus establishing a Yau–Tian–Donaldson type correspondence for the existence of Kähler metrics of constant weighted scalar curvature in the rational admissible Kähler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.

The first-named author was supported in part by an NSERC Discovery Grant.

The third-named author was supported by grant 422410 from the Simon’s Foundation.

Received 13 August 2018

Accepted 12 September 2019

Published 30 January 2023