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2014•Geometry & TopologyOpen access

Rational curves and special metrics on twistor spaces

Misha Verbitsky

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Abstract

A Hermitian metric ! on a complex manifold is called SKT or pluriclosed if dd c ! D 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.

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A Hermitian metric ! on a complex manifold is called SKT or pluriclosed if dd c ! D 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.

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Available abstract

A Hermitian metric ! on a complex manifold is called SKT or pluriclosed if dd c ! D 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.

Key concepts: Twistor theory, Twistor space, Mathematics, Pure mathematics, Moduli space, Hermitian manifold, Hermitian matrix, Space (punctuation)

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