Let $X$ be a compact complex manifold. For $p,q\ge 0$, let $A^{p,q}(X)$ denote the complex [vector space](/page/Vector%20Space) of smooth complex-valued differential forms of type $(p,q)$ on $X$, and set $A^{r,s}(X)=0$ whenever $r<0$ or $s<0$. Let $A^k(X;\mathbb C)=\bigoplus_{p+q=k}A^{p,q}(X)$, and let $d=\partial+\bar\partial$ be the [type decomposition of the exterior derivative](/theorems/7004). Assume $X$ satisfies the $\partial\bar\partial$ lemma in the following form: if $\eta\in A^{p,q}(X)$ is $d$-closed and is $\partial$-exact, $\bar\partial$-exact, or $d$-exact, then there exists $\theta\in A^{p-1,q-1}(X)$ such that $\eta=\partial\bar\partial\theta$. Then for every $k\ge 0$, every Dolbeault class in $H^{p,q}_{\bar\partial}(X)$ with $p+q=k$ admits a representative $\alpha\in A^{p,q}(X)$ satisfying $d\alpha=0$. Sending such a class $[\alpha]_{\bar\partial}$ to the de Rham class $[\alpha]_{dR}\in H^k_{dR}(X;\mathbb C)$ is independent of the chosen $d$-closed representative, and the induced map

\begin{align*} \Phi_k:\bigoplus_{p+q=k} H^{p,q}_{\bar{\partial}}(X) \longrightarrow H^k_{dR}(X;\mathbb C) \end{align*}

is an isomorphism.