Volume 18, issue 2 (2014)

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Rational curves and special metrics on twistor spaces

Misha Verbitsky

Geometry & Topology 18 (2014) 897–909
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 $ℂ\phantom{\rule{0.3em}{0ex}}{P}^{3}$ 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.

Keywords
twistor space, pluriclosed metric, K3 surface, SKT metric, rational connected variety, Moishezon variety, non-Kähler manifold
Mathematical Subject Classification 2010
Primary: 53C28
Secondary: 32Q15, 53C26