Self-dual Einstein Hermitian four manifolds

Abstract

We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperkählerian structure compatible with the negative orientation. We finally show that self-dual Einstein 4-manifolds obtained as quaternionic quotients of the Wolf spaces $\mathbb{H}{P}^2$, $\mathbb{H}{H}^2$, $SU(4)/S(U(2)U(2))$, and $SU(2,2)/S(U(2)U(2))$ are always Hermitian.

More typos corrected, 39 pages