Conformally Einstein anti-self-dual spaces and their generalisations

Abstract

We study the existence of Einstein scales and compatible metrics in Grassmannian geometry. As a special case, this includes the conformal-to-Einstein problem for anti-self-dual (ASD) conformal structures. In the ASD setting, given a genericity condition algebraic in the Weyl curvature, we obtain necessary and sufficient conditions for the existence of a local Einstein scale that refine those previously appearing in the literature. We also prove that in Riemannian signature, a non-Ricci-flat ASD-K\"ahler metric is locally conformally Einstein if and only if it has a holomorphic Killing vector field with anti-self-dual derivative proportional to the Ricci spinor. Next, we consider some generalisations of the ASD conformal-to-Einstein problem for $(2n,2)$-Grassmannian structures. When $n > 1$, given an analogous genericity condition, we obtain necessary and sufficient conditions for the existence of a local Einstein scale. Using tractor calculus, we obtain algebraic obstructions to the existence of compatible metrics and show that the unique submaximally symmetric model of the geometry has the submaximal number of linearly independent compatible metrics.