Geometry of twistor spaces

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1. We have solved a conjecture of Joyce to the effect that (S^1 × S^1)-invariant self-dual metrics are essentially those constructed by Joyce, by determining the structure of the associated twistor space in detail.2. For a twistor space Z with algebraic dimension a(Z) = 2 we have shown that either M 〜 mP^2 or M 〜 (S^1 × S^3)#mP^2 for some of its finite unramified covering M^^〜. A similar result is obtained also for the case a(Z) = 1 and of +-type.3. For a twistor space Z with a(Z) = 2 corresponding to 4P^2 we have done a detailed study of the structure of its algebraic reduction. Honda obtained significant results on the structure and existence of twistor spaces admitting an S^1-action (not necessarily semifree) in the case M = 3P^2, 4P^2.4. For any m【greater than or equal】5, by considering the deformations of Joyce twistor spaces we have obtained the first examples of a twistor space with a = 2. When m = 4 the construction gives also examples which are different from those constructed by Campana-Kreussler.5. The Case M = (S^1 × S^3)#mP^2 of +-type: When m = 0 we have completely determined the structure of algebraic reduction of Z. Whe m > 0, we have solved affirmatively the a(Z) = 1-conjecture by LeBrun for the LeBrun families which are the unique explicit example in this case. Meantime, we get a detailed description of the structure of Z.