$\overline\partial$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds
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| Format: | Preprint |
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2021
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| author | Tardini, Nicoletta Tomassini, Adriano |
| author_facet | Tardini, Nicoletta Tomassini, Adriano |
| contents | Let $(X,J)$ be a $4$-dimensional compact almost-complex manifold and let $g$ be a Hermitian metric on $(X,J)$. Denote by $Δ_{\overline\partial}:=\overline\partial\overline\partial^*+\overline\partial^*\overline\partial$ the $\overline\partial$-Laplacian. If $g$ is \emph{globally conformally Kähler}, respectively \emph{(strictly) locally conformally Kähler}, we prove that the dimension of the space of $\overline\partial$-harmonic $(1,1)$-forms on $X$, denoted as $h^{1,1}_{\overline\partial}$, is a topological invariant given by $b_-+1$, respectively $b_-$. As an application, we provide a one-parameter family of almost-Hermitian structures on the Kodaira-Thurston manifold for which such a dimension is $b_-$. This gives a positive answer to a question raised by T. Holt and W. Zhang. Furthermore, the previous example shows that $h^{1,1}_{\overline\partial}$ depends on the metric, answering to a Kodaira and Spencer's problem. Notice that such almost-complex manifolds admit both almost-Kähler and (strictly) locally conformally Kähler metrics and this fact cannot occur on compact complex manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_10594 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $\overline\partial$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds Tardini, Nicoletta Tomassini, Adriano Differential Geometry Let $(X,J)$ be a $4$-dimensional compact almost-complex manifold and let $g$ be a Hermitian metric on $(X,J)$. Denote by $Δ_{\overline\partial}:=\overline\partial\overline\partial^*+\overline\partial^*\overline\partial$ the $\overline\partial$-Laplacian. If $g$ is \emph{globally conformally Kähler}, respectively \emph{(strictly) locally conformally Kähler}, we prove that the dimension of the space of $\overline\partial$-harmonic $(1,1)$-forms on $X$, denoted as $h^{1,1}_{\overline\partial}$, is a topological invariant given by $b_-+1$, respectively $b_-$. As an application, we provide a one-parameter family of almost-Hermitian structures on the Kodaira-Thurston manifold for which such a dimension is $b_-$. This gives a positive answer to a question raised by T. Holt and W. Zhang. Furthermore, the previous example shows that $h^{1,1}_{\overline\partial}$ depends on the metric, answering to a Kodaira and Spencer's problem. Notice that such almost-complex manifolds admit both almost-Kähler and (strictly) locally conformally Kähler metrics and this fact cannot occur on compact complex manifolds. |
| title | $\overline\partial$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2104.10594 |