Kählerity of Einstein four-manifolds

We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemannian cover if necessary) Kähler–Einstein, provided that \(\lambda _2 \ge -\frac{S}{12}\) ${\lambda }_2\ge -\frac{S}{12}$ , where \(\lambda _2\) ${\lambda }_2$ is the middle eigenvalue of the self-dual Weyl tensor \(W^+\) $W^{+}$ and S is the scalar curvature. An analogous result holds for closed oriented four-manifolds with \(\delta W^+=0\) $\delta W^{+}=0$ .