Dolbeault-Poincaré lemma

On a sufficiently small polydisc in a complex manifold, every $\bar{\partial }$-closed $(p,q)$-form with $q>0$ is locally $\bar{\partial }$-exact. A coefficientwise one-variable integral operator and induction over the coordinates prove the result.

On a disc $D\subset \mathbb{C}$, the Cauchy-Green operatoris a right inverse to $\partial /\partial \bar{z}$ in the interior: ${\partial }_{\bar{z}}Tg=g$. Applying it to coefficients gives the local homotopy operator used in the Dolbeault-Poincaré lemma.