Einstein Metrics on Complex Surfaces

Suppose M a compact manifold which admits an Einstein metric g which is Kähler with respect to some complex structure J . Is every other Einstein metric h on M also Kähler-Einstein? If the complex dimension of (M,J) is ≥ 3, the answer is generally no; for example, CP3 admits both the FubiniStudy metric, which is Kähler-Einstein, and a non-Kähler Einstein metric [2] obtained by appropriately squashing the fibers of the twistor projection CP3 → S. Iterated Cartesian products with CP1 then provide counterexamples in all higher dimensions. However, if M is a 4-manifold, so that (M,J) is a compact complex surface, there is reason to hope that the anwer to the above question might be yes. Indeed, Hitchin [12] was able to answer the question in the affirmative for complex surfaces which admit Ricci-flat Kähler metrics; his argument hinges on the fact that any 4-dimensional Einstein manifold satisfies