Dolbeault resolution of holomorphic differential forms ($0\to {\Omega }^p\to {\mathcal{A}}^{p,0}\stackrel{\bar{\partial }}{\to }{\mathcal{A}}^{p,1}\to \cdots$)

The Dolbeault-Poincaré lemma makes this a locally exact complex of sheaves, with degree-zero kernel the holomorphic differential forms. The sheaves of smooth forms are fine sheaves by a partition of unity. The acyclic resolution theorem therefore computes the cohomology of ${\Omega }^p$ by the global Dolbeault operator complex.