An ansatz for almost-Kahler, Einstein 4-manifolds

We describe an ansatz due to Tod which describes all known 4-dimensional strictly almost-Kahler, Einstein manifolds. We show that all weakly *-Einstein, strictly almost-Kahler Einstein 4-manifolds arise from this ansatz. We apply the ideas developed in the proof of this result to show that constant curvature manifolds of any dimension do not admit strictly almost-Kahler structures. In dimensions 6 this result is new. In dimensions 8 and above it reproves a result of [13].