On Einstein, Hermitian 4-manifolds

Abstract

Let $(M, h)$ be a compact 4-dimensional Einstein manifold, and suppose that $h$ is Hermitian with respect to some complex structure $J$ on $M$. Then either $(M, J, h)$ is Kähler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on $\mathbb{CP}_2\#\overline{\mathbb{CP}}_2$, or the Einstein metric on $\mathbb{CP}_2\#2\overline{\mathbb{CP}}_2$ discovered in "On conformally Kähler, Einstein manifolds," J. Amer. Math. Soc. 21 (2008), pp. 1137–1168.