I am wondering if the following argument is true:


Let $X$ be a $\dim n$ compact projective complex manifold, let $\alpha\in H^{2n-2}(X,\mathbb{Q})$ be a cohomology class. If for any ample line bundle $L$, we have $c\_1(L)\cup \alpha=0$, can I argue that $\alpha=0$?



Cup Product with Ample Line Bundles
No. If $\alpha $ is of type $(n-2,n)$, its product with any class of type $(1,1)$ is zero, but $\alpha $ is not necessarily zero (you can take for instance $\alpha = c\_1(L)^{n-2}[\omega ]$, where $L$ is an ample line bundle and $\omega $ a nonzero holomorphic 2-form).


