Geometry of twistor space

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1. We have obtained an affimative answer to a conjugate of Joyce to the effect that a simply connected and complete self-dual manifold which admits a smooth action of a 2-torus is diffeomorphic to a connected sum mCP (2) of m copies of complex projective plane and its self-dual structure is isomorphic to one of the examples constructed by Joyce himself in 1995. Our method was to use the twistor space associated to the given self-dual manifold. In fact, more generally, without simple-connectivity assumption we have classified compact self-dual manifolds which admit an action of a two torus. As a byproduct of these investigation we could determine very precise structure as a complex manifold of the twistor space associated to Joyce self-dual metric. In particular we have shown that there exists a nice bimeromorphic model of the twistor space which is realized as a fiber space of torus surfaces and that this latter structure is completely determined by the invariant of the original smooth action of the torus on mCP (2). The twistor space is a Moishezon manifold and it is interesting to construct a natural birational projecitve model.2. We have studied the deformation space of Joyce twistor space and have shown that its Kuranishi space is nonsingular. Furthermore, using such deformation we have given, for every integer m【greater than or equal】4 the first examples of self-dual structure of positive type on mCP (2) whose twistor space has algebraic dimension two. This result determines distribution of algebraic dimensions of twistor spaces associated to mCP (2). Furthermore, in the case of 4CP (2) we have given a detailed description of the twistor space as a general elliptic fiber space and then by taking its branched covering constructed a interesting family of compact complex manifolds whose canonical bundle is trivial.