Kaehler geometry of black holes and gravitational instantons

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.