This is probably a silly question but I wasn't able to find a quick answer. let $\omega$ be a smooth $(p,q)$-form on a complex manifold $X$ of dimension $d$. It generates a current of integration $T_\omega$ as follows. For any compactly supported and smooth $(d-p,n-q)$-form $\eta$ we define $$T_\omega:\eta\mapsto\int_X\omega\wedge\eta$$
Now suppose that $T_\omega$ is constantly $0$, is it true that $\omega=0$?