Rational curves and special metrics on twistor spaces

Verbitsky, Misha¹

¹ Faculty of Mathematics, National Research University Higher School of Economics, Laboratory of Algebraic Geometry, 7 Vavilova Str., Moscow 117312, Russia

Geometry & topology, Tome 18 (2014) no. 2, pp. 897-909.

Résumé

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.

Détail

DOI : 10.2140/gt.2014.18.897

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

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