Spaces of complex null geodesics
L. J. Mason, L. P. Hughston, P. Z. Kobak, K. Pulverer · 2022
Spaces of complex null geodesies give one of the natural routes for the attempt to generalize the nonlinear graviton construction to general space-times. The basic construction (see §III.2.2 for details) works in the holomorphic category, but otherwise in full generality: one starts with a complex manifold M n with holomorphic conformai structure [ g ] and a complex conformai equivalence class of connections [∇] possibly with torsion so that one can define holomorphic complex torsion null geodesies in M n . Perhaps after restriction to a suitable open set in M n , the space of complex null geodesies, P N, is a complex manifold of dimension 2n − 3 and this plays the role of the twistor space. As in the nonlinear graviton, a point of M corresponds to a compact holomorphic submanifold Q of P N, where Q is now a projective quadric of dimension n − 2. In §III.2.2 , LeBrun shows that, for n > 3, M n can be reconstructed as (perhaps an open subset of) the moduli space of embeddings of Q into P . The incidence properties of the Q ’s in P then determine [ g ] and [∇]. The correspondence is preserved under small deformations.