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 CP2≠CP2¯, or the Einstein metric on CP2≠2CP2¯ discovered in [8]. © 2012 J. Differential Geometry.