Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913882834993152 |
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| author | Huang, Teng Wang, Weiwei |
| author_facet | Huang, Teng Wang, Weiwei |
| contents | In this article, we discuss the spaces of harmonic forms $\mathcal{H}^{\bullet}_{d}$ over a closed almost Kähler manifold $(X, J,ω)$. We show that if the almost complex structure $J$ on the almost Kähler manifold $X$ is not too non-integrable in some sense, then the spaces $\mathcal{H}^{\bullet}_{d}$ have the Hodge decomposition $\mathcal{H}^{k}_{d}=\oplus_{p+q=k}\mathcal{H}^{p,q}_{d}$. As a consequence, the not too non-integrable almost complex structure $J$ is complex $C^{\infty}$-pure-and-full, and the Hard Lefschetz Condition (HLC) on $\mathcal{H}^{\bullet}_{d}$ is satisfied. Moreover, we can prove a rigidity result for the closed $4$-dimensional almost Kähler manifold with $b^{+}_{2}(X)\geq2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds Huang, Teng Wang, Weiwei Differential Geometry In this article, we discuss the spaces of harmonic forms $\mathcal{H}^{\bullet}_{d}$ over a closed almost Kähler manifold $(X, J,ω)$. We show that if the almost complex structure $J$ on the almost Kähler manifold $X$ is not too non-integrable in some sense, then the spaces $\mathcal{H}^{\bullet}_{d}$ have the Hodge decomposition $\mathcal{H}^{k}_{d}=\oplus_{p+q=k}\mathcal{H}^{p,q}_{d}$. As a consequence, the not too non-integrable almost complex structure $J$ is complex $C^{\infty}$-pure-and-full, and the Hard Lefschetz Condition (HLC) on $\mathcal{H}^{\bullet}_{d}$ is satisfied. Moreover, we can prove a rigidity result for the closed $4$-dimensional almost Kähler manifold with $b^{+}_{2}(X)\geq2$. |
| title | Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.06402 |