Hard Lefschetz property for $\mathbb{S}^3$-actions

TL;DR

This paper extends the Hard Lefschetz Property (HLP) to almost-free S3-actions on manifolds, proving equivalence of two versions under certain conditions and confirming HLP in 3-Sasakian manifolds and specific examples.

Contribution

It introduces two versions of HLP for S3-actions, proves their equivalence under a cohomological condition, and demonstrates HLP in 3-Sasakian and certain free actions.

Findings

HLP holds for 3-Sasakian manifolds.

Two versions of HLP are equivalent under a cohomological condition.

Examples of free S3-actions satisfying HLP but not being 3-Sasakian.

Abstract

The Hard Lefschetz Property (HLP) has recently been formulated in the context of isometric flows without singularities on manifolds. In this category, two versions of the HLP (transverse and not) have been proven to be equivalent, thus generalizing what happens in the important cases of both K-contact and Sasakian manifolds. In this work we define both versions of the HLP for almost-free S3 -actions, and prove that they agree for actions satisfying a cohomological condition, which includes the important category of 3-Sasakian manifolds, where those two versions of the HLP are shown to be held. We also provide a family of examples of free actions of the 3-sphere which are not 3-Sasakian manifolds, but satisfy the HLP.