TITLE: Higher degree generalizations of the Hard Lefschetz Theorem
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: I guess I knew that I should've looked in part 2 of Lazarsfeld's book before posting this question, but it was checked out of the library... 
Anyway, the speculation in the last paragraph is correct: if $X$ is a smooth projective variety and $E$ is an ample vector bundle on $X$ of rank $e$, then $c_e(E)$ has the $2e$-HLP. This is exactly the statement of Theorem 7.1.10 in Positivity in Algebraic Geometry II. (Lazarsfeld only considers multiplication by a single top Chern class, not powers of it, but this is equivalent by replacing $E$ by $E^{\oplus r}$).