Compositio Mathematica

On Some Twistor Spaces Over $4{\Bbb CP}$2

Nobuhiro Honda a1
a1 Department of Mathematics, Faculty of Science, Hiroshima University, 739-8526, Japan. E-mail:

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We show that for any positive integer τ there exist on $4{\Bbb CP}$2, the connected sum of four complex projective planes, twistor spaces whose algebraic dimensions are two. Here, τ appears as the order of the normal bundle of C in S, where S is a real smooth half-anti-canonical divisor on the twistor space and C is a real smooth anti-canonical divisor on S. This completely answers the problem posed by Campana and Kreussler. Our proof is based on the method developed by Honda, which can be regarded as a generalization of the theory of Donaldson and Friedman.

Key Words: twistor space; self-dual metric; algebraic dimension; elliptic curve.