Dolbeault theorem

For the sheaf ${\Omega }^p$ of holomorphic $p$-forms on a complex manifold, the Dolbeault-Poincaré lemma makesa resolution by fine sheaves. Taking global sections therefore gives the natural isomorphism

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.