Let $\alpha$ and $\beta$ be two complex $(1,1)$ forms defined as:
$\alpha = \alpha_{ij} dx^i \wedge d\bar x^j$
$\beta= \beta_{ij} dx^i \wedge d\bar x^j$
Let's say, I know the following:
1) $\alpha \wedge \beta = 0$
2) $\beta \neq 0$
I want to somehow show that the only way to achieve (1) is by forcing $\alpha = 0$. Are there general known conditions on the $\beta_{ij}$ for this to happen?
The only condition I could think of is if all the $\beta_{ij}$ are the same. However, this is a bit too restrictive. I'm also interested in the above problem when $\beta$ is a $(2,2)$ form.