Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry

Peng Wu [22] recently announced a beautiful characterization of conformally Kähler, Einstein metrics of positive scalar curvature on compact oriented 4-manifolds via the condition det(W ⁺) > 0. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences of our method, which builds on techniques previously developed in [16].