Let $X$ be a complex manifold of complex dimension $n$, and let $p$ be an integer with $0 \leq p \leq n$. Let $\Omega_X^p$ denote the sheaf of holomorphic $p$-forms on $X$, and let $\mathcal A_X^{p,q}$ denote the sheaf of smooth complex-valued $(p,q)$-forms on $X$. Then the sequence of sheaves