A Hermitian metric ω on a complex manifold is called SKT or pluriclosed if dd~c ω = 0. Let M be a twistor space of a compact, anti-self dual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kahler, hence isomorphic to CP~3 or a flag space. This result is obtained from rational connectedness of the twistor space, due to F Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein.
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