Higher dimensional generalizations of twistor spaces

We construct a generalization of twistor spaces of hypercomplex manifolds and hyper-Kahler manifolds $M$, by generalizing the twistor ${\mathbb{P}}^1$ to a more general complex manifold $Q$. The resulting manifold $X$ is complex if and only if $Q$ admits a holomorphic map to ${\mathbb{P}}^1$. We make branched double covers of these manifolds. Some class of these branched double covers can give rise to non-Kahler Calabi-Yau manifolds. We show that these manifolds $X$ and their branched double covers are non-Kahler. In the cases that $Q$ is a balanced manifold, the resulting manifold $X$ and its special branched double cover have balanced Hermitian metrics.