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 $\beta \leadsto \beta_1 = \beta +df$, 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 $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_\beta)$ 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 Ra\'{z}ny \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 $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_\beta)$ are topological invariants of $X$.