A Kodaira type conjecture on almost complex 4 manifolds
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916225506869248 |
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| author | Lin, Dexie |
| author_facet | Lin, Dexie |
| contents | Not long ago, Cirici and Wilson defined a Dolbeault cohomology on almost complex manifolds to answer Hirzebruch's problem. In this paper, we define a refined Dolbeault cohomology on almost complex manifolds. We show that the condition $\tilde h^{1,0}=\tilde h^{0,1}$ implies a symplectic structure on a compact almost complex $4$ manifold, where $\tilde h^{1,0}$ and $\tilde h^{0,1}$ are the dimensions of the refined Dolbeault cohomology groups with bi-degrees $(1,0)$ and $(0,1)$ respectively. Combining the partial answer to Donaldson's tameness conjecture, we offer a sufficient condition for a compact almost complex $4$ manifold to become an almost Kähler one.
Moreover, we prove that the condition $\tilde{h}^{1,0}=\tilde h^{0,1}$ is equivalent to the generalized $\partial\bar\partial$-lemma. This can be regarded as an analogue of the Kodaira's conjecture on almost complex $4$ manifolds. As an application, we show that the Kodaira-Thurston manifold satisfies the $\partial\bar\partial$-lemma. Meanwhile, we show that the Frölicher-type equality does not hold on a general almost complex $4$ manifold, which is different to the case of compact complex surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14690 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Kodaira type conjecture on almost complex 4 manifolds Lin, Dexie Differential Geometry Geometric Topology Symplectic Geometry 57R30, 53A45, 53C23, 53D35 Not long ago, Cirici and Wilson defined a Dolbeault cohomology on almost complex manifolds to answer Hirzebruch's problem. In this paper, we define a refined Dolbeault cohomology on almost complex manifolds. We show that the condition $\tilde h^{1,0}=\tilde h^{0,1}$ implies a symplectic structure on a compact almost complex $4$ manifold, where $\tilde h^{1,0}$ and $\tilde h^{0,1}$ are the dimensions of the refined Dolbeault cohomology groups with bi-degrees $(1,0)$ and $(0,1)$ respectively. Combining the partial answer to Donaldson's tameness conjecture, we offer a sufficient condition for a compact almost complex $4$ manifold to become an almost Kähler one. Moreover, we prove that the condition $\tilde{h}^{1,0}=\tilde h^{0,1}$ is equivalent to the generalized $\partial\bar\partial$-lemma. This can be regarded as an analogue of the Kodaira's conjecture on almost complex $4$ manifolds. As an application, we show that the Kodaira-Thurston manifold satisfies the $\partial\bar\partial$-lemma. Meanwhile, we show that the Frölicher-type equality does not hold on a general almost complex $4$ manifold, which is different to the case of compact complex surfaces. |
| title | A Kodaira type conjecture on almost complex 4 manifolds |
| topic | Differential Geometry Geometric Topology Symplectic Geometry 57R30, 53A45, 53C23, 53D35 |
| url | https://arxiv.org/abs/2307.14690 |