In a [paper][1] of Derdzinski<sup>1</sup> (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.

<sup>1</sup>Derdziński, Andrzej: *Self-dual Kähler manifolds and Einstein manifolds of dimension four*, Compositio Mathematica, Volume 49 (1983) no. 3 , p. 405-433.


  [1]: http://www.numdam.org/item?id=CM_1983__49_3_405_0