Rational curves and special metrics on twistor spaces

Verbitsky, Misha

doi:10.2140/gt.2014.18.897

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Rational curves and special metrics on twistor spaces

Misha Verbitsky

Geometry & Topology 18 (2014) 897–909

DOI: 10.2140/gt.2014.18.897

Abstract

A Hermitian metric $ω$ on a complex manifold is called SKT or pluriclosed if $d d^{c} ω = 0$ . 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 $ℂ P^{3}$ 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 $K 3$ 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

References

Publication

Received: 27 October 2012

Revised: 24 August 2013

Accepted: 1 November 2013

Published: 7 April 2014

Proposed: Gang Tian

Seconded: Richard Thomas, Yasha Eliashberg

Authors

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

Volume 18, issue 2 (2014)