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arXiv · 2509.06664

Hodge theoretic results for nearly K\"ahler manifolds in all dimensions

Michael Albanese·Spiro Karigiannis·Lucía Martín-Merchán·Aleksandar Milivojević
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Abstract

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.

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Michael Albanese, Spiro Karigiannis, Lucía Martín-Merchán, Aleksandar Milivojević. 2025-09-08. Hodge theoretic results for nearly K\"ahler manifolds in all dimensions. https://doi.org/10.4310/ajm.260406230501

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