Holomorphic de Rham complex (${\Omega }_M^{\bullet }$)

This complex has the sheaves ${\Omega }_M^p$ of holomorphic differential forms and differential $\partial$. By the holomorphic Poincaré lemma, it resolves the complex constant sheaf.

The sheaf is ${\mathcal{Z}}_M^1=\ker (\partial :{\Omega }_M^1\to {\Omega }_M^2)$. The holomorphic Poincaré lemma gives $0\to \underline{\mathbb{C}}\to {\mathcal{O}}_M\stackrel{\partial }{\to }{\mathcal{Z}}_M^1\to 0$. On a compact complex surface its long exact sequence in sheaf cohomology yields $0\to H^0(M,{\Omega }_M^1)\to H^1(M,\mathbb{C})\to H^1(M,{\mathcal{O}}_M)$.

On a star-shaped holomorphic coordinate neighbourhood, a closed holomorphic differential form of positive degree has a holomorphic primitive. Radial integration gives the homotopy operator. This proves exactness of the holomorphic de Rham complex on stalks.