Self-dual Einstein Hermitian four manifolds

TL;DR

This paper classifies self-dual Einstein four-manifolds with Hermitian structures, characterizes those with hyperhermitian structures, and shows certain quaternionic quotients are always Hermitian, advancing understanding of their geometric properties.

Contribution

It provides a local classification of self-dual Einstein Hermitian four-manifolds and characterizes hyperhermitian structures compatible with negative orientation.

Findings

Self-dual Einstein Hermitian four-manifolds are classified locally.

Hyperhermitian, non-hyperkählerian structures are characterized.

Quaternionic quotients of Wolf spaces are always Hermitian.

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-hyperk\"ahlerian 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.