In a paper of Derdzinski (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 metric itself is Kahler.

I was wondering are there further characterization of Kahler metrics on four-manifolds? Thank you very much.