Four--Dimensional Metrics Conformal to Kahler

We derive some necessary conditions on a Riemannian metric $(M, g)$ in four dimensions for it to be locally conformal to K\"ahler. 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 K\"ahler metric in the conformal class. In the anti--self--dual case we establish a one to one correspondence between K\"ahler 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 K\"ahler. We establish a link between the `conformal to K\"ahler 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 K\"ahler metric or a para-K\"ahler metric.