In algebraic geometry, the cohomology groups $H^{\bullet}(\mathbb{P}^n,\mathcal{O}(k))$ are well-understood. As far as I know, in most textbooks these cohomology groups are computed via Cech cohomology or Koszul resolutions.
My question is: in the case of $\mathbb{CP}^{n}$, could we compute $H^{\bullet}(\mathbb{CP}^n,\mathcal{O}(k))$ analytically via Dolbeault cohomology? Is there any references？