In a paper of Derdzinski1 (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the scalar curvature or $|W^+|^2$ is constant, then the metric itself is Kahler.
I was wondering are there further characterization of Kahler metrics on four-manifolds? Thank you very much.
1Derdziński, Andrzej: Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Mathematica, Volume 49 (1983) no. 3 , p. 405-433.
