Existence of twistor spaces of algebraic dimension two over the connected sum of four complex projective planes

Campana, F.; Kreußler, B.

doi:10.1090/S0002-9939-99-05406-4

Existence of twistor spaces of algebraic dimension two over the connected sum of four complex projective planes
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by F. Campana and B. Kreußler PDF

Proc. Amer. Math. Soc. 127 (1999), 2633-2642 Request permission

Abstract:

We prove the existence of twistor spaces of algebraic dimension two over the connected sum of four complex projective planes $4 \mathbb {C}\mathbb {P}^2$. These are the first examples of twistor spaces of algebraic dimension two over a simply connected Riemannian four–manifold with positive scalar curvature. For this purpose we develop a method to distinguish between twistor spaces of algebraic dimension one and two by looking at the order of a certain point in the Picard group of an elliptic curve.

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