Volume 18, issue 2 (2014)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Rational curves and special metrics on twistor spaces

Misha Verbitsky

Geometry & Topology 18 (2014) 897–909
Abstract

A Hermitian metric ω on a complex manifold is called SKT or pluriclosed if ddcω = 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 P3 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
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