Four--Dimensional Metrics Conformal to Kahler

TL;DR

This paper investigates conditions under which four-dimensional Riemannian metrics are conformal to Kähler metrics, providing necessary and sufficient criteria, characterizations, and examples, and linking to projective structures in two dimensions.

Contribution

It establishes new necessary and sufficient conditions for four-dimensional metrics to be conformal to Kähler, including characterizations in anti-self-dual cases and connections to projective structures.

Findings

Characterization of conformal to Kähler metrics in four dimensions.

Existence of ASD metrics not conformal to Kähler.

Link between conformal to Kähler condition and projective metrisability.

Abstract

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.