On the Weyl tensor of a self-dual complex 4-manifold

摘要

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also show that the projective structure of the beta-surfaces of a self-dual manifold is flat. All these results are illustrated in detail in the case of the complexification of CP2. [References: 23]