Invariant nearly-Kähler structures

This paper considers invariant almost Hermitian structures on a flag manifold $G/P=U/K$ where $G$ is a complex semi-simple Lie group, $P$ is a parabolic subgroup of $G$, $U$ is a compact real form of $G$ and $K=U\cap P$ is the centralizer of a torus. The main result shows that there are nearly-K\"{a}hler structures in $G/P$ which are not K\"{a}hler if and only if $G/P$ has height three. This proves for the flag manifolds a conjecture by J.A. Wolf and A. Gray.