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

TL;DR

This paper studies the basic de Rham cohomology associated with Reeb flows on contact manifolds, establishing invariance under certain deformations and linking it to topological properties via Hodge structures and Lefschetz properties.

Contribution

It demonstrates the invariance of basic de Rham complexes under specific contact form deformations and connects the cohomology to topological invariants through Hodge structures and Lefschetz properties.

Findings

Basic forms vanish when contracted with Reeb vector field.

Basic de Rham cohomology is invariant under certain contact form deformations.

On closed manifolds, basic cohomology reflects topological invariants.

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 $X$.