Hodge theoretic results for nearly Kähler manifolds in all dimensions

Abstract

We generalize to nearly Kähler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly Kähler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly Kähler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact Kähler 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-Kähler manifolds, which are a particular class of $(4n+2)$-dimensional nearly Kähler manifolds equipped with a special $\mathrm{SU}(n)\cdot \mathrm{U}(1)$ structure.

15 pages. Version 2: very minor revisions following referee's reports. Final version, to appear in Asian Journal of Mathematics