Conformally K\"ahler, Einstein-Maxwell Geometry

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}.