Bott-Chern Poincaré lemma

On a polydisc, $H_{BC}^{p,q}=0$ for $p,q>0$. A Poincare lemma primitive for a closed pure-type form can be modified by exact terms until only bidegrees $(p-1,q)$ and $(p,q-1)$ remain. The Dolbeault-Poincaré lemma and conjugate Dolbeault-Poincaré lemma then make the original form $\partial \bar{\partial }$-exact.