Contact de Rham Cohomology and Hodge Structures Transversal to Reeb Foliations
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mathematics-14-01450-v2.pdf
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439.42 KB
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aa93d228679972da39473c12653b7a11
Author(s)
Katz, Gabriel
Date Issued
April 25, 2026
Journal
Mathematics
Publisher
MDPI
Citation
Katz, G. Contact de Rham Cohomology and Hodge Structures Transversal to Reeb Foliations. Mathematics 2026, 14, 1450.
Version
Final published version
Abstract
Let β be a contact form on a compact smooth manifold X and vβ its Reeb vector field. This study applies the general results of different authors regarding Hodge structures that are transversal to a given foliation to the special case of 1-dimensional foliation generated by the Reeb flow vβ. The de Rham differential complex Ω∗ basic(X, vβ) of so-called basic forms relative to vβ-flow differential forms is the focus of this investigation. By definition, basic forms vanish when being contracted with vβ, and so do their differentials. We prove that under the change of β ⇝ β1 = β + d f , where a function f : X → R such that d f(vβ) > −1, the differential complexes Ω∗ basic(X, vβ1 ) and Ω∗ basic(X, vβ) are canonically isomorphic. We investigate when the 2-form dβ and its powers deliver nontrivial elements in the basic de Rham cohomology H∗ basic dR (X, vβ) of the differential complex Ω∗ basic(X, vβ). Answers to these questions contrast sharply in the cases of a closed X and an X with boundary. Building on the work of Ra´zny, we show that on a closed manifold X equipped with a transversal to the Reeb flow Hodge structure that satisfies the Basic Hard Lefschetz Property, the basic de Rham cohomology H∗ basic dR (X, vβ) is a topological invariant of X.
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DOI of Published Version
https://doi.org/10.3390/math14091450
