The complex Goldberg-Sachs theorem in higher dimensions

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We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+\epsilon)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $\epsilon \in {0,1}$ and $m \geq 2$.

• In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarrantee the integrability of $\mathcal{N}$, and in odd dimensions, of its orthogonal complement.
• These results generalise the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor, and the complex Goldberg-Sachs theorem from four to higher dimensions.
• Higher-dimensional analogues of the Petrov type D condition are defined, and we show that these lead to the integrability of up to $2^m$ holomorphic distributions of totally null $m$-planes.