Rational curves and special metrics on twistor spaces

Abstract

A Hermitian metric $\omega$ on a complex manifold is called SKT or pluriclosed if $d{d}^c\omega =0$. 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 Kähler, 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.