Hard Lefschetz Property for Isometric Flows
José Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, Robert Wolak
TL;DR
The paper extends the Hard Lefschetz Property to isometric flows by defining transversal $(THL)_k$ and global $(HL)_k$ dualities tied to the Euler class $[e]$ and Lefschetz operator $L(\cdot)=\cdot\wedge e$, and proves their equivalence in dimension $2n+1$ using primitive basic cohomology and the Gysin sequence. It provides constructive tools, notably the Kobayashi S$^1$-bundle construction to realize Lefschetz isometric flows with prescribed Euler classes, and presents a spectrum of examples outside the Sasakian/K-contact setting, including Lefschetz but non-contact flows and non-Lefschetz counterexamples. The results clarify how HLP phenomena extend beyond transversely Kähler and Sasakian geometries, offering a topological, metric-independent criterion for Lefschetz behavior in isometric flows and enabling systematic exploration of which categories admit Lefschetz dualities. The work thus broadens the applicability of Lefschetz-type dualities in foliated manifolds and provides explicit, topologically robust examples for classification and SEO.
Abstract
The Hard Lefschetz Property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), but a new version of the HLP has been recently given in terms of duality of the cohomology of the manifold itself in arXiv:1306.2896. Both properties were proved to be equivalent (see arXiv:1311.1431) in the case of K-contact flows. In this paper we extend both versions of the HLP (transverse and not) to the more general category of isometric flows, and show that they are equivalent. We also give some explicit examples which illustrate the categories where the HLP could be considered.