We generalize to nearly K\"ahler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly K\"ahler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly K\"ahler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact K\"ahler manifold. In the $6$-dimensional case, Verbitsky was able to say slightly more using the induced $\mathrm{ SU}(3)$ structure. We discuss potential extensions of this to twistor spaces over positive scalar curvature quaternionic-K\"ahler manifolds, which are a particular class of $(4n+2)$-dimensional nearly K\"ahler manifolds equipped with a special $\mathrm{ SU}(n) \! \cdot \! \mathrm{ U}(1)$ structure.
Categories: math.DG