Contact de Rham cohomology and Hodge structures transversal to the Reeb foliations

Abstract

Let $\beta$ be a contact form on a compact smooth manifold $X$ and $v_{\beta }$ 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 $v_{\beta }$. The de Rham differential complex ${\Omega }_{\mathsf{basic}}^{\ast }(X,v_{\beta })$ of, so called, {\sf basic} relative to $v_{\beta }$-flow differential forms is in the focus of this investigation. By definition, the basic forms vanish when being contracted with $v_{\beta }$, and so do their differentials. We prove that under the change $\beta ⇝{\beta }_1=\beta +df$, where a function $f:X\to \mathbf{R}$ such that $df(v_{\beta })>-1$, the differential complexes ${\Omega }_{\mathsf{basic}}^{\ast }(X,v_{{\beta }_1})$ and ${\Omega }_{\mathsf{basic}}^{\ast }(X,v_{\beta })$ are canonically isomorphic. We investigate when the $2$-form $d\beta$ and its powers deliver nontrivial elements in the basic de Rham cohomology $H_{\mathsf{basic}d\mathcal{R}}^{\ast }(X,v_{\beta })$ of the differential complex ${\Omega }_{\mathsf{basic}}^{\ast }(X,v_{\beta })$. 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 $H_{\mathsf{basic}d\mathcal{R}}^{\ast }(X,v_{\beta })$ are topological invariants of $X$.

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