Conformally Kähler structures

We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non–linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to ${\mathrm{\Lambda }}^3(\text{cT})$, where $\text{cT}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms. Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing-Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterisation of algebraically special Riemannian metrics of type $D$ in dimensions higher than four.