The Einstein–Maxwell equations, Kähler metrics, and Hermitian geometry

Any constant-scalar-curvature Kähler (cscK) metric on a complex surface may be viewed as a solution of the Einstein–Maxwell equations, and this allows one (LeBrun, 2010; Shu, 2009) to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b1b1 even. However, not all solutions of the Einstein–Maxwell equations on such manifolds arise in this way; new examples can be constructed by means of conformally Kähler geometry.