Let $X$ be a complex manifold. For each integer $k \geq 0$, let $A^k(X)$ denote the complex [vector space](/page/Vector%20Space) of smooth complex-valued differential $k$-forms on $X$. For each pair of integers $p,q \geq 0$, let $A^{p,q}(X)$ denote the complex vector space of smooth complex-valued differential forms of type $(p,q)$ on $X$, so that the type decomposition is

\begin{align*} A^k(X)=\bigoplus_{a+b=k} A^{a,b}(X). \end{align*}

Let

\begin{align*} d: A^k(X) \to A^{k+1}(X) \end{align*}

denote the [exterior derivative](/theorems/1525) on smooth complex-valued forms. Since $X$ is a complex manifold, its complex structure is integrable, and hence for every $p,q \geq 0$ the exterior derivative maps $A^{p,q}(X)$ into $A^{p+1,q}(X) \oplus A^{p,q+1}(X)$. Define

\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*}

as the $(p+1,q)$ and $(p,q+1)$ type components of $d$ on $A^{p,q}(X)$. Then, as operators on smooth complex-valued differential forms on $X$,

\begin{align*} \partial^2 = 0, \qquad \bar{\partial}^2 = 0, \qquad \partial \bar{\partial} + \bar{\partial}\partial = 0. \end{align*}

Equivalently, for every $p,q \geq 0$ and every $\alpha \in A^{p,q}(X)$,

\begin{align*} \partial(\partial \alpha)=0, \qquad \bar{\partial}(\bar{\partial}\alpha)=0, \qquad \partial(\bar{\partial}\alpha)+\bar{\partial}(\partial\alpha)=0. \end{align*}