Spaces of complex null geodesics in complex-Riemannian geometry

The notion of a complex - Riemannian n n - manifold, meaning a complex n 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 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 geodesies. This relationship between local geometry and global complex analysis is stable under deformations.