Let $X$ be a compact complex manifold of complex dimension $n$ satisfying the $\partial\bar\partial$ lemma. For each pair of integers $p,q$, let $A^{p,q}(X)$ denote the complex [vector space](/page/Vector%20Space) of smooth complex-valued differential forms of type $(p,q)$ on $X$, with $A^{p,q}(X)=0$ if $p<0$, $q<0$, $p>n$, or $q>n$. Let $A^k(X;\mathbb C)=\bigoplus_{p+q=k}A^{p,q}(X)$ and $A^\bullet(X;\mathbb C)=\bigoplus_{k\geq 0}A^k(X;\mathbb C)$. Let $d:A^k(X;\mathbb C)\to A^{k+1}(X;\mathbb C)$ be the [exterior derivative](/theorems/1525), and let $d=\partial+\bar\partial$ be its type decomposition, where $\partial:A^{p,q}(X)\to A^{p+1,q}(X)$ and $\bar\partial:A^{p,q}(X)\to A^{p,q+1}(X)$. Assume the $\partial\bar\partial$ lemma in the following form: every smooth complex-valued form that is $d$-exact, $\partial$-closed, and $\bar\partial$-closed is $\partial\bar\partial$-exact, equivalently componentwise in pure bidegree. Then the differential graded algebra $(A^\bullet(X;\mathbb C),d,\wedge)$ is formal over $\mathbb C$; equivalently, it is connected to its cohomology algebra $(H^\bullet(X;\mathbb C),0,\smile)$ by a zigzag of differential graded algebra quasi-isomorphisms over $\mathbb C$.