Summary: We derive some necessary conditions on a Riemannian metric \((M, g)\) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti-self-dual, the self-dual Weyl spinor must be of algebraic type \(D\) and satisfy a simple first order conformally invariant condition which is necessary and sufficient for the existence of a Kähler metric in the conformal class. In the anti-self-dual case we establish a one to one correspondence between Kähler metrics in the conformal class and non-zero parallel sections of a certain connection on a natural rank ten vector bundle over \(M\). We use this characterisation to provide examples of ASD metrics which are not conformal to Kähler.
We establish a link between the ‘conformal to Kähler condition’ in dimension four and the metrisability of projective structures in dimension two. A projective structure on a surface \(U\) is metrisable if and only if the induced (2,2) conformal structure on \(M = TU\) admits a Kähler metric or a para-Kähler metric.