A new construction of anti-self-dual four-manifolds

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D 2 V = 0 on a five-dimensional manifold . The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in { V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space has , projecting thus over with twistor lines mapping onto plane conics.