Rational curves and special metrics on twistor spaces

A Hermitian metric omega on a complex manifold is called SKT or pluriclosed if dd(c)omega = 0. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kahler, hence isomorphic to CP3 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.