Let $X$ be a complex manifold of complex dimension $n$. For $0 \leq p,q \leq n$, let $A^{p,q}(X)$ denote the space of smooth complex-valued differential forms of type $(p,q)$ on $X$, and let
be the space of smooth complex-valued differential forms on $X$. Let $\partial$ and $\bar\partial$ be the Dolbeault components of the [exterior derivative](/theorems/1525) $d$, so that on smooth complex-valued forms
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\begin{align*}
d = \partial + \bar\partial,
\end{align*}
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with $\partial: A^{r,s}(X) \to A^{r+1,s}(X)$ and $\bar\partial: A^{r,s}(X) \to A^{r,s+1}(X)$, extended linearly to $A^\bullet(X)$.
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If $\alpha \in A^{p,q}(X)$ and $\beta \in A^\bullet(X)$, then