Homogeneous Einstein metrics on non-K\"ahler C-spaces

We study homogeneous Einstein metrics on indecomposable non-K\"ahlerian C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank} G>\mathsf{rank} H$ over flag manifolds $F=G/K$ of a compact simple Lie group $G$. Based on the theory of painted Dynkin diagrams we present the classification of such spaces. Next we focus on the family $$M_{\ell, m, n}:=\mathsf{SU}(\ell+m+n)/\mathsf{SU}(\ell)\times\mathsf{SU}(m)\times\mathsf{SU}(n)\,,\quad \ell, m, n\in\mathbb{Z}_{+}$$ and examine several of its geometric properties. We show that invariant metrics on $M_{\ell, m, n}$ are not diagonal and beyond certain exceptions their parametrization depends on six real parameters. By using such an invariant Riemannian metric, we compute the diagonal and the non-diagonal part of the Ricci tensor and present explicitly the algebraic system of the homogeneous Einstein equation. For general positive integers $\ell, m, n$, by applying mapping degree theory we provide the existence of at least one $\mathsf{SU}(\ell+m+n)$-invariant Einstein metric on $M_{\ell, m, n}$. For $\ell=m$ we show the existence of two $\mathsf{SU}(2m+n)$ invariant Einstein metrics on $M_{m, m, n}$, and for $\ell=m=n$ we obtain four $\mathsf{SU}(3n)$-invariant Einstein metrics on $M_{n, n, n}$. We also examine the isometry problem for these metrics, while for a plethora of cases induced by fixed $\ell, m, n$, we provide the numerical form of all non-isometric invariant Einstein metrics.