Killing–Yano tensors and multi-Hermitian structures

We show that the Euclidean Kerr–NUT-(A)dS metric in 2m2m dimensions locally admits 2m2m Hermitian complex structures. These are derived from the existence of a non-degenerate closed conformal Killing–Yano tensor with distinct eigenvalues. More generally, a conformal Killing–Yano tensor, provided its exterior derivative satisfies a certain condition, algebraically determines 2m2m almost complex structures that turn out to be integrable as a consequence of the conformal Killing–Yano equations. In the complexification, these lead to 2m2m maximal isotropic foliations of the manifold and, in Lorentz signature, these lead to two congruences of null geodesics. These are not shear-free, but satisfy a weaker condition that also generalises the shear-free condition from four dimensions to higher dimensions. In odd dimensions, a conformal Killing–Yano tensor leads to similar integrable distributions in the complexification. We show that the recently discovered five-dimensional solution of Lü, Mei and Pope also admits such integrable distributions, although this does not quite fit into the story as the obvious associated two-form is not conformal Killing–Yano. We give conditions on the Weyl curvature tensor imposed by the existence of a non-degenerate conformal Killing–Yano tensor; these give an appropriate generalisation of the type D condition on a Weyl tensor from four dimensions.