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Abstract

We obtain a closed formula for the Kaehler potential of a broad class of
four-dimensional Lorentzian or Euclidean conformal "Kaehler" geometries,
including the Plebanski-Demianski class and various gravitational instantons
such as Fubini-Study and Chen-Teo. We show that the Kaehler potentials of
Schwarzschild and Kerr are related by a Newman-Janis shift. Our method also
shows that a class of supergravity black holes, including the Kerr-Sen
spacetime, is Hermitian (but not conformal Kaehler). We finally show that the
integrability conditions of complex structures lead naturally to the
(non-linear) Weyl double copy, and we give new vacuum and non-vacuum examples
of this relation.