Rational curves and special metrics on twistor spaces

TL;DR

The paper proves that twistor spaces of anti-selfdual manifolds with pluriclosed metrics are Kähler, leading to classification results, and also shows the moduli space of rational curves on certain twistor spaces is Stein.

Contribution

It establishes that such twistor spaces are Kähler and classifies them as either  or flag spaces, using rational connectedness and properties of rational curves.

Findings

Twistor spaces with pluriclosed metrics are Kähler.

Such twistor spaces are isomorphic to  or flag spaces.

The moduli space of rational curves on K3 twistor spaces is Stein.

Abstract

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 K\"ahler, hence isomorphic to $\C P^3$ 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.