Spaces of complex null geodesics in complex-riemannian geometry

The notion of a complex-Riemannian n-manifold, meaning a complex n-manifold with a nondegenerate complex quadratic form on each tangent space which varies holomorphically from point to point, is briefly developed. It is shown that, provided n ≥ 4, every such manifold locally arises canonically as the moduli space of all quadrics of a fixed normal-bundle type in an associated space of complex null geodesics. This relationship between local geometry and global complex analysis is stable under deformations.