In a [paper][1] 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 the metric itself is Kahler.

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


  [1]: http://www.numdam.org.proxy.library.cornell.edu/item?id=CM_1983__49_3_405_0