Hodge theoretic results for nearly Kähler manifolds in all dimensions
Michael Albanese, Spiro Karigiannis, Lucía Martín-Merchán, Aleksandar Milivojević
TL;DR
This work extends Verbitsky's Hodge-theoretic framework from strict nearly Kähler 6-manifolds to NK manifolds in all dimensions, building an intrinsic operator approach on the exterior algebra and deriving a generalized set of Kähler-like identities. It proves a dimension-agnostic Hodge decomposition for Δ_d and shows that, on compact NK manifolds, the associated Hodge numbers h^{p,q} satisfy h^{p,q} = h^{q,p} and the Betti numbers b^k decompose accordingly, with strong vanishing constraints in the 6-dimensional strict case. The results reveal that many hallmark Kähler features persist beyond dimension six, though certain SU(3)-specific refinements remain dimension-dependent; the paper also discusses potential extensions to related geometric structures, such as twistor spaces over quaternionic-Kähler bases. These findings point toward broader applicability of nearly Kähler identities, potential ∂∂̄-lemma analogues, and new routes to formality in a wider geometric setting.
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.