Weighted extremal K\"ahler metrics and the Einstein-Maxwell geometry of projective bundles

TL;DR

This paper establishes existence criteria for weighted extremal K"ahler metrics on admissible projective bundles, leading to new conformally K"ahler Einstein-Maxwell examples and linking metric existence to a weighted K-stability condition.

Contribution

It provides a general existence theorem for weighted extremal metrics on admissible bundles, extending Calabi's extremal metric theory and constructing higher-dimensional Einstein-Maxwell metrics.

Findings

Existence of weighted extremal K"ahler metrics depends on an explicit function’s positivity.

Many new conformally K"ahler Einstein-Maxwell metrics are constructed in higher dimensions.

A weighted K-stability condition is established, linking geometric stability to metric existence.

Abstract

We study the existence of weighted extremal K\"ahler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge K\"ahler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-T{\o}nnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal K\"ahler metrics pioneered by a seminal construction of Calabi. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally K\"ahler, Einstein-Maxwell metrics of complex dimension $m>2$, thus extending the recent constructions of LeBrun and Koca-T{\o}nnesen-Friedman to higher dimensions. For each admissible K\"ahler 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 K\"ahler metric in the given class, whereas if it is strictly negative somewhere in $(-1,1)$, there is no K\"ahler 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 K\"ahler metrics of constant weighted scalar curvature in the rational admissible K\"ahler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.