Let $X$ be a complex manifold. For each pair $p,q\ge 0$, let $A^{p,q}(X)$ denote the space of smooth complex-valued differential forms of type $(p,q)$ on $X$, and let $A^\bullet(X)=\bigoplus_{k\ge 0}A^k(X;\mathbb C)$ denote the graded complex [vector space](/page/Vector%20Space) of all smooth complex-valued differential forms on $X$. Let $d:A^\bullet(X)\to A^{\bullet+1}(X)$ be the [exterior derivative](/theorems/1525), and let $\partial$ and $\bar\partial$ be the Dolbeault operators characterized on forms of type $(p,q)$ by

\begin{align*} \partial:A^{p,q}(X)\to A^{p+1,q}(X). \end{align*}

and

\begin{align*} \bar\partial:A^{p,q}(X)\to A^{p,q+1}(X). \end{align*}

With the convention that $d^c:A^\bullet(X)\to A^{\bullet+1}(X)$ is the complex-linear operator

\begin{align*} d^c=i(\bar\partial-\partial), \end{align*}

one has the identity of complex-linear differential operators on $A^\bullet(X)$

\begin{align*} dd^c=2i\,\partial\bar\partial. \end{align*}