Killing-Yano tensors and multi-hermitian structures

Understand

We show that the Euclidean Kerr-NUT-(A)dS metric in $2m$ dimensions locally admits $2^m$ 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 $2^m$ almost complex structures that turn out to be integrable as a consequence of the conformal Killing-Yano equations.
• In the complexification, these lead to $2^m$ maximal isotropic foliations of the manifold and, in Lorentz signature, these lead to two congruences of null geodesics.