Hard Lefschetz property for $\mathbb{S}^3$-actions
JosÉ Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, Robert Wolak
TL;DR
This work extends the Hard Lefschetz Property (HLP) to almost-free $ abla$-actions of the 3-sphere by defining transverse (THL) and global (HL) versions on basic and total cohomology and establishing when they coincide. The authors introduce an injectivity condition on multiplication by the Euler class $oldsymbol{ abla}$ in basic cohomology and prove THL$_{}$ $ ext{iff}$ HL$_{}$ under this condition, thereby giving a precise equivalence criterion for Lefschetz actions. They prove that 3-Sasakian manifolds satisfy the injectivity condition and hence fulfill both THL and HL, verifying the HLP in this important class. Additionally, they construct families of almost-free $ abla$-actions on manifolds of dimension $4n+3$ that satisfy HLP without admitting a compatible 3-Sasakian structure, illustrating that HLP is broader than 3-Sasakian geometry and can occur in new topological settings.
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.