I am trying at the moment to understand Schubert calculus, and have taken the simple example of the complex projective line ${\mathbb CP}^1$ as a guide. Now in the simplest formulation I know, we have that $$ \Omega^0({\mathbb CP}^1) \simeq \Omega^2({\mathbb CP}^1) \simeq {\cal O}({\mathbb CP}^1), $$ and $$ \Omega^{(0,1)}({\mathbb CP}^1) \simeq L_{-2}, ~~~ \Omega^{(1,0)}({\mathbb CP}^1) \simeq L_{2}, $$ where $L_{-2}$ and $L_{2}$, are the vector bundles corresponding to $-2$ and $2$ in the standard classification of the line bundles over ${\mathbb CP}^{1}$ in terms of ${\mathbb Z}$.

Can anyone give me a concrete presentation of the cohomology groups $H^{p,q}$ in terms of this description? Moreover, what are the Schubert cell generators of the groups?

A concrete presentation of how all this works for ${\mathbb CP}^2$ and ${\mathbb CP}^3$ would also be very welcome.