On Some Twistor Spaces Over 4ℂℙ²

We show that for any positive integer τ there exist on 4ℂℙ², 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.