Kählerian Twistor Spaces

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The work of R. Penrose has shown the deep relationship between the conformal structure of Minkowski space and the complex geometry of lines in projective three-space. This transformation applies to a more general class of conformal structures and in particular there is a riemannian version, described in [5], which is itself of interest to mathematical physicists. Basically, to each oriented riemannian 4manifold X one associates canonically a 6-manifold Z, the twistor space of X, together with an almost complex structure. If this structure is integrable, then Z becomes a complex 3-manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold X. It is natural to ask what sort of complex manifold one obtains by this process. In this paper we ask the more specific question: which compact 4c-manifolds have Ka'hlerian twistor spaces' 1 . The answer, rather surprisingly, is that there are just two examples: the 4-sphere S* and the complex projective plane P2{ C), each with their standard homogeneous conformal structures. The twistor space of S 4 is the 3-dimensional projective space P3(C), and that of P 2(Q is the manifold F2{C) of flags in C 3 . The essential point of the proof of this result is the remark that if Z is a twistor space with a Kahler metric, then the canonical bundle K is negative. Using vanishing theorems and the Riemann-Roch theorem we then narrow down the possibilities for Z to four cases: projective space, the flag manifold, the intersection of two quadrics in P5, and the double covering of P3 branched over a non-singular quartic surface. The last two candidates cannot be realized, however, because the value of their Euler characteristic is incompatible with their being twistor spaces. Throughout the proof we make essential use of a real structure on Z, which reduces considerably the occurrence of singularities at various stages. Aside from this, our methods are the standard ones of algebraic geometry. The four examples to which the proof leads us are all Fano threefolds. If we drop the requirement of compactness on X, then it is quite possible for an open set of a Fano threefold to be a twistor space. In particular, we show that the family of conies lying on the intersection of two