Contact de Rham cohomology and Hodge structures transversal to the Reeb foliations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912844754190336 |
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| author | Katz, Gabriel |
| author_facet | Katz, Gabriel |
| contents | Let $β$ be a contact form on a compact smooth manifold $X$ and $v_β$ its Reeb vector field. The paper applies general results of different authors about Hodge structures that are transversal to a given foliation to the special case of $1$-dimensional foliation generated by the Reeb flow $v_β$.
The de Rham differential complex $Ω_{\mathsf{basic}}^\ast(X, v_β)$ of, so called, {\sf basic} relative to $v_β$-flow differential forms is in the focus of this investigation. By definition, the basic forms vanish when being contracted with $v_β$, and so do their differentials.
We prove that under the change $β\leadsto β_1 = β+df$, where a function $f:X \to \mathbf R$ such that $df(v_β) > -1$, the differential complexes $Ω_{\mathsf {basic}}^\ast(X, v_{β_1})$ and $Ω_{\mathsf{basic}}^\ast(X, v_β)$ are canonically isomorphic.
We investigate when the $2$-form $dβ$ and its powers deliver nontrivial elements in the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ of the differential complex $Ω_{\mathsf{basic}}^\ast(X, v_β)$. Answers to these questions contrast sharply in the cases of a closed $X$ and a $X$ with boundary.
On the other hand, building on work of Raźny \cite{Raz}, we show that on a closed manifold $X$, equipped with a transversal to the Reeb flow Hodge structure that satisfies the {\it Basic Hard Lefschetz Property}, the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ are topological invariants of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Contact de Rham cohomology and Hodge structures transversal to the Reeb foliations Katz, Gabriel Differential Geometry Dynamical Systems Let $β$ be a contact form on a compact smooth manifold $X$ and $v_β$ its Reeb vector field. The paper applies general results of different authors about Hodge structures that are transversal to a given foliation to the special case of $1$-dimensional foliation generated by the Reeb flow $v_β$. The de Rham differential complex $Ω_{\mathsf{basic}}^\ast(X, v_β)$ of, so called, {\sf basic} relative to $v_β$-flow differential forms is in the focus of this investigation. By definition, the basic forms vanish when being contracted with $v_β$, and so do their differentials. We prove that under the change $β\leadsto β_1 = β+df$, where a function $f:X \to \mathbf R$ such that $df(v_β) > -1$, the differential complexes $Ω_{\mathsf {basic}}^\ast(X, v_{β_1})$ and $Ω_{\mathsf{basic}}^\ast(X, v_β)$ are canonically isomorphic. We investigate when the $2$-form $dβ$ and its powers deliver nontrivial elements in the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ of the differential complex $Ω_{\mathsf{basic}}^\ast(X, v_β)$. Answers to these questions contrast sharply in the cases of a closed $X$ and a $X$ with boundary. On the other hand, building on work of Raźny \cite{Raz}, we show that on a closed manifold $X$, equipped with a transversal to the Reeb flow Hodge structure that satisfies the {\it Basic Hard Lefschetz Property}, the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ are topological invariants of $X$. |
| title | Contact de Rham cohomology and Hodge structures transversal to the Reeb foliations |
| topic | Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2502.09773 |