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# Reference for when a metric on a four-manifold is Kahler?

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 the metric itself is Kahler.

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

user38600
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