Conformally K\"ahler, Einstein-Maxwell Geometry

TL;DR

This paper investigates the existence of special conformally K"ahler metrics with constant scalar curvature and Ricci tensor of type (1,1), linking geometric analysis with stability conditions and providing new explicit examples in four dimensions.

Contribution

It introduces a momentum map framework and stability criteria for conformally K"ahler, Einstein-Maxwell metrics, extending previous work and providing explicit constructions in four-dimensional toric orbifolds.

Findings

Established a Futaki invariant as an obstruction to existence.

Defined K-polystability as a necessary condition for toric cases.

Proved sufficiency of K-polystability on certain 4-orbifolds with second Betti number 2.

Abstract

On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of K\"ahler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In real dimension $4$, such Hermitian metrics provide a Riemannian counter-part of the Einstein--Maxwell (EM) equations in general relativity, and have been recently studied in \cite{ambitoric1, LeB0, LeB, KTF}. We show how the existence problem of such Hermitian metrics (which we call in any dimension {\it conformally K\"ahler, EM} metrics) fits into a formal momentum map interpretation, analogous to results by Donaldson and Fujiki~\cite{donaldson, fujiki} in the cscK case. This leads to a suitable notion of a Futaki invariant which provides an obstruction to the existence of conformally K\"ahler, EM metrics invariant under a certain group of automorphisms which are associated to a given K\"ahler class, a real holomorphic vector field on $(M,J)$, and a positive normalization constant. Specializing to the toric case, we further define a suitable notion of $K$-polystability and show it provides a (stronger) necessary condition for the existence of toric, conformally K\"ahler, EM metrics. We use the methods of \cite{ambitoric2} to show that on a compact symplectic toric $4$-orbifold with second Betti number equal to $2$, $K$-polystability is also a sufficient condition for the existence of (toric) conformally K\"ahler, EM metrics, and the latter are explicitly described as ambitoric in the sense of \cite{ambitoric1}. As an application, we exhibit many new examples of conformally K\"ahler, EM metrics defined on compact $4$-orbifolds, and obtain a uniqueness result for the construction in \cite{LeB0}.