  The author has elsewhere given a complete classification of those compact
oriented Einstein 4-manifolds on which the self-dual Weyl curvature is
everywhere positive in the direction of some self-dual harmonic 2-form. In this
article, similar results are obtained when the self-dual Weyl curvature is
everywhere non-negative in the direction of a self-dual harmonic 2-form that is
transverse to the zero section of the bundle of self-dual 2-forms. However,
this transversality condition plays an essential role in the story; dropping it
leads one into wildly different territory where entirely different phenomena
predominate.
