Holomorphic Poincaré lemma

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.

For a closed holomorphic one-form $\sum b_{k}d{z}_k$ on a polydisc, the displayed coordinate-path integral is a holomorphic function. Differentiating in $z_j$ gives the $k=j$ endpoint term; for $k>j$ use ${\partial }_j{b}_k={\partial }_k{b}_j$ and integrate that derivative. The resulting sum telescopes to $b_j(z)$. Thus $\partial F=\sum b_{k}d{z}_k$, giving an explicit degree-one proof of the holomorphic Poincaré lemma.