Hodge theoretic results for nearly Kähler manifolds in all dimensions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911568092987392 |
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| author | Albanese, Michael Karigiannis, Spiro Martín-Merchán, Lucía Milivojević, Aleksandar |
| author_facet | Albanese, Michael Karigiannis, Spiro Martín-Merchán, Lucía Milivojević, Aleksandar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06664 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge theoretic results for nearly Kähler manifolds in all dimensions Albanese, Michael Karigiannis, Spiro Martín-Merchán, Lucía Milivojević, Aleksandar Differential Geometry 53C25, 53C26 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. |
| title | Hodge theoretic results for nearly Kähler manifolds in all dimensions |
| topic | Differential Geometry 53C25, 53C26 |
| url | https://arxiv.org/abs/2509.06664 |