Let $M$ be a $2d$-dimensional manifold. We say that $\omega \in H^2(M)$ has the Hard Lefschetz Property (HLP) if multiplication with $\omega^j$ is an isomorphism $H^{d-j} \to H^{d+j}$. This holds for the first Chern class of an ample line bundle on a smooth projective variety, this is the classical Hard Lefschetz Theorem.

My question is whether there is a generalization to classes in $H^4, H^6,$ ... etc.

My proposed definition is that $\eta \in H^k(M)$ has the k-HLP if multiplication by $\eta^j$ from $H^i$ to $H^{i+jk}$ is invective (surjective) if $|i-d|$ is greater (smaller) than $|i+jk-d|$. Then 2-HLP is HLP and if $\omega$ has the HLP then $\omega^j$ has the $2j$-HLP.

Are there other examples of classes with the $2j$-HLP? E.g. Chern classes of vector bundles satisfying an appropriate ampleness condition?