On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-Kähler structures
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| Format: | Preprint |
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2022
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| _version_ | 1866911174549831680 |
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| author | Rao, Sheng Zhang, Runze |
| author_facet | Rao, Sheng Zhang, Runze |
| contents | Inspired by a recent work of D. Wei--S. Zhu on the extension of closed complex differential forms and Voisin's usage of the $\partial\bar{\partial}$-lemma, we obtain several new theorems of deformation invariance of Hodge numbers and reprove the local stabilities of $p$-Kähler structures with the $\partial\bar{\partial}$-property. Our approach is more concerned with the $d$-closed extension by means of the exponential operator $e^{ι_φ}$.
Furthermore, we prove the local stabilities of transversely $p$-Kähler structures with mild $\partial\bar{\partial}$-property by adapting the power series method to the foliated case, which strengthens the works of A. El Kacimi Alaoui--B. Gmira and P. Raźny on that of the transversely Kähler foliations with homologically orientability. We observe that a transversely Kähler foliation, even without homologically orientability, also satisfies the $\partial\bar{\partial}$-property. So even when $p=1$ (transversely Kähler), our results are new as we can drop the assumption in question on the initial foliation. Several theorems on the deformation invariance of basic Hodge/Bott--Chern numbers with mild $\partial\bar{\partial}$-properties are also presented. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_06870 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-Kähler structures Rao, Sheng Zhang, Runze Complex Variables Differential Geometry 32G05 (primary), 13D10, 14D15, 53C55, 53C12 (secondary) Inspired by a recent work of D. Wei--S. Zhu on the extension of closed complex differential forms and Voisin's usage of the $\partial\bar{\partial}$-lemma, we obtain several new theorems of deformation invariance of Hodge numbers and reprove the local stabilities of $p$-Kähler structures with the $\partial\bar{\partial}$-property. Our approach is more concerned with the $d$-closed extension by means of the exponential operator $e^{ι_φ}$. Furthermore, we prove the local stabilities of transversely $p$-Kähler structures with mild $\partial\bar{\partial}$-property by adapting the power series method to the foliated case, which strengthens the works of A. El Kacimi Alaoui--B. Gmira and P. Raźny on that of the transversely Kähler foliations with homologically orientability. We observe that a transversely Kähler foliation, even without homologically orientability, also satisfies the $\partial\bar{\partial}$-property. So even when $p=1$ (transversely Kähler), our results are new as we can drop the assumption in question on the initial foliation. Several theorems on the deformation invariance of basic Hodge/Bott--Chern numbers with mild $\partial\bar{\partial}$-properties are also presented. |
| title | On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-Kähler structures |
| topic | Complex Variables Differential Geometry 32G05 (primary), 13D10, 14D15, 53C55, 53C12 (secondary) |
| url | https://arxiv.org/abs/2204.06870 |