Anti-self-dual conformal structures with null Killing vectors from projective structures

Abstract

Using twistor methods, we explicitly construct all local forms of four--dimensional real analytic neutral signature anti--self--dual conformal structures $(M,[g])$ with a null conformal Killing vector. We show that $M$ is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of $(M,[g])$ by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface orthogonal in $(M, [g])$. We give examples of conformal classes which contain Ricci--flat metrics on compact complex surfaces and discuss other conformal classes with no Ricci--flat metrics.Comment: 43 pages, 4 figures. Theorem 2 has been improved: ASD metrics are given in terms of general projective structures without needing to choose special representatives of the projective connection. More examples (primary Kodaira surface, neutral Fefferman structure) have been included. Algebraic type of the Weyl tensor has been clarified. Final version, to appear in Commun Math Phy