  Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may
be viewed as a solution of the Einstein-Maxwell equations, and this allows one
to produce solutions of these equations on any 4-manifold that arises as a
compact complex surface with b_1 even. It is shown, however, that not all
solutions of the Einstein-Maxwell equations on such manifolds arise in this
way; new examples can be constructed by means of conformally Kaehler geometry.
