Why Complex projective spaces don't admit some differential forms?
To be more specific, I know that the space of complex forms is decomposed as direct sum of holomorphic and anti-holomorphic part; given that, I don't understand why the complex projective line $CP^1$ for instance has only the scalar 0-form , and $dz \wedge \bar{dz}$.
Why 1-forms like $dz$ and $\bar{dz}$ vanish?
It's not that $dz$ vanishes, it's that $dz$ has a pole of order $2$ at infinity in the projective line ($w = 1/z$ is a holomorphic coordinate in a neighborhood of infinity, and $dw = -dz/z^{2}$), so $dz$ is a meromorphic $1$-form, but not holomorphic. Similar remarks hold for higher dimensional projective spaces.