Compatible almost complex structures on the Hard Lefschetz condition
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929729624342528 |
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| author | Lin, Dexie |
| author_facet | Lin, Dexie |
| contents | For a compact Kähler manifold, it is well-established that its de Rham cohomology satisfies the Hard Lefschetz condition, which is reflected in the equality between the Betti numbers and the Hodge numbers. A special subclass of symplectic manifolds also adheres to this condition. Cirici and Wilson \cite{CW20} employ the variant Hodge number to propose a sufficient criterion for compact almost Kähler manifolds to satisfy this condition. In this paper, we show that this condition is only sufficient by presenting examples of compact almost Kähler manifolds that fulfill the Hard Lefschetz condition while violating the equality between the variant Hodge numbers and Betti numbers, that is, \[b^1>2h^{1,0}.\] This phenomenon contrasts with the behavior observed in compact Kähler manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_16948 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Compatible almost complex structures on the Hard Lefschetz condition Lin, Dexie Differential Geometry Symplectic Geometry 32A25, 53C56, 53D35 For a compact Kähler manifold, it is well-established that its de Rham cohomology satisfies the Hard Lefschetz condition, which is reflected in the equality between the Betti numbers and the Hodge numbers. A special subclass of symplectic manifolds also adheres to this condition. Cirici and Wilson \cite{CW20} employ the variant Hodge number to propose a sufficient criterion for compact almost Kähler manifolds to satisfy this condition. In this paper, we show that this condition is only sufficient by presenting examples of compact almost Kähler manifolds that fulfill the Hard Lefschetz condition while violating the equality between the variant Hodge numbers and Betti numbers, that is, \[b^1>2h^{1,0}.\] This phenomenon contrasts with the behavior observed in compact Kähler manifolds. |
| title | Compatible almost complex structures on the Hard Lefschetz condition |
| topic | Differential Geometry Symplectic Geometry 32A25, 53C56, 53D35 |
| url | https://arxiv.org/abs/2312.16948 |