Einstein Metrics, Harmonic Forms and Conformally Kähler Geometry

The author has elsewhere given a complete classification of the compact oriented Einstein 4-manifolds that satisfy W⁺(ω, ω) > 0 for some self-dual harmonic 2-form ω, where W⁺ denotes the self-dual Weyl curvature. In this chapter, similar results are obtained when W+(ω, ω) ≥ 0, provided the self-dual harmonic 2-form ω is transverse to the zero section of Λ⁺ → M. However, this transversality condition plays an essential role in the story; dropping it leads one into wildly different territory where entirely different phenomena predominate.