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

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

2006-01-17

arxiv.org/abs/math/0601419v4

Commun.Math.Phys.272:85-118,2007

Mathematics

Differential Geometry

43 pages, 4 figures. Theorem 2 has been improved: ASD metrics are given in terms of general projective structures without need

Scientific paper

10.1007/s00220-007-0208-4

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.

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