Contact de Rham cohomology and Hodge structures transversal to the Reeb foliations
Gabriel Katz
TL;DR
The paper addresses transversal Hodge structures to Reeb foliations on compact (including boundary) manifolds by formulating and studying the basic de Rham complex $\Omega^*_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ and its cohomology $H^*_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ relative to the Reeb flow $v_β$. It establishes deformation and isotopy invariance of these basic complexes, connects the basic differential $dβ$ to Lyapunov functions and boundary phenomena, and develops an odd-dimensional Hodge framework on the transverse bundle $ξ_β$ including a Basic Hard Lefschetz property and Lefschetz decomposition. The results show that when the Basic Hard Lefschetz property holds for the Reeb foliations, the basic cohomology is invariant under changes of the contact form (and under Lefschetz-equivalent correspondences), and they relate these invariants to trajectory data and asymptotic cycles. The work also conjectures trajectory-space–dependent behavior of basic cohomology for traversing and boundary-generic flows, aiming to fill gaps for manifolds with boundary and to connect dynamics, foliation theory, and transverse Hodge theory in contact geometry.
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 $v_β$. Theses applications are available for the Reeb flows on {\sf closed} manifolds $X$. In contrast, for the Reeb flows on manifolds with boundary, little is known about the Hodge structures transversal to the $v_β$-flow. We are trying to fill in this gap.\smallskip The de Rham differential complex $Ω_{\mathsf b}^\ast(X, v_β)$ 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. In particular, 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 $Ω_{\mathsf b}^\ast(X, v_β)$. Answers to these questions seem to contrast sharply the cases of a closed $X$ and a $X$ with boundary. %we prove that when a $v_β$-flow admits a Lyapunov function, then the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ of the complex $Ω_{\mathsf b}^\ast(X, v_β)$ are topological invariants of $X$. On the other hand, building on work of Raźny \cite{Raz}, we show that on closed manifolds, equipped with a transversal to the Reeb flow Hodge structure that satisfies the {\sf Basic Hard Lefschetz property}, the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_β)$ are topological invariants of $X$.