Kählerity of Einstein four-manifolds

Abstract: We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemann cover if necessary) K\"ahler-Einstein, provided that $\lambda_2 \geq -\frac{S}{12}$, where $\lambda_2$ is the middle eigenvalue of the self-dual Weyl tensor $W<sup>+$ and $S$ is the scalar curvature. The same conclusion holds for closed oriented four-manifolds with $\delta W<sup>+=0$.