Conformally Kähler, Einstein–Maxwell geometry

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 Kähler 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 equations in general relativity, and have been recently studied in [3, 34, 35, 33]. We show how the existence problem of such Hermitian metrics (which we call in any dimension conformally Kähler, Einstein–Maxwell metrics) fits into a formal momentum map interpretation, analogous to results by Donaldson and Fujiki [22, 25] in the constant scalar curvature Kähler case. This leads to a suitable notion of a Futaki invariant which provides an obstruction to the existence of conformally Kähler, Einstein–Maxwell metrics invariant under a certain group of automorphisms which are associated to a given Kähler 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 Kähler, Einstein–Maxwell metrics. We use the methods of [4] 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 Kähler, Einstein–Maxwell metrics, and the latter are explicitly described as ambitoric in the sense of [3]. As an application, we exhibit many new examples of conformally Kähler, Einstein–Maxwell metrics defined on compact 4-orbifolds, and obtain a uniqueness result for the construction in [34].