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

Abstract

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 $X$.