Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $\alpha \in H^{2k}(X)$ be an algebraic cohomology class. Let us fix an $l$ such that $H^l(X) \neq 0$ and $H^{l+2k}(X) \neq 0$.

The general question (perhaps a bit vague) is: does anybody know a sufficient condition on $X$ such that the intersection map

$ \alpha \cup - : H^l(X) \to H^{l+2k}(X) $

is NOT the zero map?

In particular, I would be interested in one of the following two cases.

1) Both $h^l$ and $h^{l+2k}$ can be arbitrarily large (so that being the zero map is very unlikely, so to speak).

2) The number $2l+2k$ equals $2 \dim_{\mathbb{C}}(X)$, thus $H^l\cong H^{l+2k}$ by Poincarè duality.

Would it help to assume that $\alpha$ is the Euler class (top Chern class) of a vector bundle on $X$?

ADDED: I am interested in the case when $l$ is odd.