I try to compute the de Rham cohomology of the complex projective spaces $\mathbb{C}\mathbb{P}^n$ without using Mayer-Vietoris sequence but only using definiton of the de Rham cohomology, i.e. the quotient vector space $\{k$-closed form$/ k$-exact form$\}$.
Firstly, I take $\{U_i\}$ as the standard affine cover of $\mathbb{C}\mathbb{P}^n$, then try to coincide $k$-form$^\ast$ on the intersections, and hope to arrive that (*)for $k$ is odd, every closed $k$-form is also exact $k$-form to conclude that cohomology group is trivial. (**)for $k$ is even, i know that from Fubini-Study form $\omega$ gives that there is an unique $2$-form on $\mathbb{C}\mathbb{P}^n$, and generates other cohomology groups. But i do not know the proof that $\omega^k$ is unique $2k$ form which generates other related cohomology group.
My question is that there is a way to do that showing all closed $k$-forms are also exact when $k$ is odd?
Would you mind if i ask you to explain?
$\ast$ where $k$-form $(p,q)$-types and $k=p+q$