Weakly positive form that is not positive

Question

Let $X$ be a complex manifold of (complex)dimension $k$ and $\alpha$ be a smooth $(p,p)$ form on $X$. Then $\alpha$ is said to be positive if for each point, there exists an open set where $\alpha$ can be written as finite sum of forms $(i\beta_{1}\wedge\bar{\beta_{1}})\wedge\ldots(i\beta_{p}\wedge\bar{\beta_{p}})$, where $\beta_{i}$ are smooth $(1,0)$ forms on that set.
On the other hand, $\alpha$ is weakly positive if $\alpha\wedge\beta$ is a positive $(k,k)$ form for any positive $(k-p,k-p)$ form $\beta$ on $X$. Now, from the definition of a positive form, it is clear that any positive form is weakly positive. But I can't think of any weakly positive form which is not positive.

Answer 1

What about something like $i(dz_1\wedge d\bar z_2 + dz_1\wedge d\bar z_1)$ in $\Bbb C^2$ (or $i(dz_1\wedge d\bar z_2 + dz_3\wedge d\bar z_3)$ in $\Bbb C^3$)?