Let $X$ be a complex manifold of complex dimension $n$. For each integer $k$ with $0 \leq k \leq 2n$, let $A^k(X)$ denote the complex [vector space](/page/Vector%20Space) of smooth complex-valued differential $k$-forms on $X$. For integers $p,q$, let $A^{p,q}(X)$ denote the complex vector space of smooth differential forms of type $(p,q)$ on $X$, with the convention that $A^{p,q}(X)=0$ if $p \notin \{0,\dots,n\}$ or $q \notin \{0,\dots,n\}$. Then

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