Let $X$ be a compact complex manifold of complex dimension $n$. Let $A^k(X;\mathbb C)$ denote the space of smooth complex-valued differential $k$-forms on $X$, with [exterior derivative](/theorems/1525) $d:A^k(X;\mathbb C)\to A^{k+1}(X;\mathbb C)$. For $0\le p,q\le n$, let $A^{p,q}(X)$ denote the space of smooth complex-valued forms of type $(p,q)$, and set $A^{p,q}(X)=0$ if either index is negative or greater than $n$. Let $\partial:A^{p,q}(X)\to A^{p+1,q}(X)$ and $\bar{\partial}:A^{p,q}(X)\to A^{p,q+1}(X)$ be the unique type components of $d$ from [citetheorem:TEMP-7], so that $d=\partial+\bar{\partial}$ on $A^{p,q}(X)$, and assume the [Dolbeault operator relations](/theorems/8046) from [citetheorem:TEMP-8]. Define the Bott-Chern cohomology group by

\begin{align*} H^{p,q}_{BC}(X):=\frac{\ker(\partial:A^{p,q}(X)\to A^{p+1,q}(X)) \cap \ker(\bar{\partial}:A^{p,q}(X)\to A^{p,q+1}(X))}{\operatorname{im}(\partial\bar{\partial}:A^{p-1,q-1}(X)\to A^{p,q}(X))}. \end{align*}

Define the Aeppli cohomology group by

\begin{align*} H^{p,q}_{A}(X):=\frac{\ker(\partial\bar{\partial}:A^{p,q}(X)\to A^{p+1,q+1}(X))}{\partial A^{p-1,q}(X)+\bar{\partial}A^{p,q-1}(X)}. \end{align*}

For each $0\le p,q\le n$, define the Bott-Chern-to-Aeppli comparison map $\iota_{p,q}:H^{p,q}_{BC}(X)\to H^{p,q}_{A}(X)$ by $[\alpha]_{BC}\mapsto [\alpha]_{A}$. For each $0\le k\le 2n$, define the Bott-Chern-to-de Rham comparison map $\Phi_k:\bigoplus_{p+q=k}H^{p,q}_{BC}(X)\to H^k_{dR}(X;\mathbb C)$ by

\begin{align*} \Phi_k\left(\sum_{p+q=k}[\alpha^{p,q}]_{BC}\right):=\left[\sum_{p+q=k}\alpha^{p,q}\right]_{dR}. \end{align*}

Here each $\alpha^{p,q}\in A^{p,q}(X)$ is a Bott-Chern representative. Use the Bott-Chern-de Rham comparison theorem in the following precise form: for this compact complex manifold $X$, the cohomological $\partial\bar{\partial}$ lemma stated in condition 1 below holds if and only if every $\Phi_k$ is an isomorphism; under this isomorphism, surjectivity gives representatives decomposed into $d$-closed pure-type Bott-Chern components, and injectivity identifies two such decompositions precisely up to a $\partial\bar{\partial}$ correction in each bidegree. The following conditions are equivalent.

1. The cohomological $\partial\bar{\partial}$ lemma holds: for every $0\le p,q\le n$ and every pure-type form $\alpha\in A^{p,q}(X)$ satisfying $\partial\alpha=0$, $\bar{\partial}\alpha=0$, and

\begin{align*} \alpha\in \partial A^{p-1,q}(X)+\bar{\partial}A^{p,q-1}(X), \end{align*}

there exists $\gamma\in A^{p-1,q-1}(X)$ such that

\begin{align*} \alpha=\partial\bar{\partial}\gamma. \end{align*}

2. For every $0\le p,q\le n$, the comparison map $\iota_{p,q}$ is injective. 3. For every $0\le k\le 2n$, the map $\Phi_k$ is an isomorphism. Equivalently, every de Rham cohomology class $[\beta]\in H^k_{dR}(X;\mathbb C)$ has a representative of the form

\begin{align*} \beta'=\sum_{p+q=k}\beta^{p,q} \end{align*}

where each $\beta^{p,q}\in A^{p,q}(X)$ is $d$-closed, and if $\sum_{p+q=k}\beta^{p,q}$ and $\sum_{p+q=k}\widetilde{\beta}^{p,q}$ are two such representatives of the same de Rham class, then for every $p+q=k$ there exists $\gamma^{p-1,q-1}\in A^{p-1,q-1}(X)$ such that

\begin{align*} \beta^{p,q}-\widetilde{\beta}^{p,q}=\partial\bar{\partial}\gamma^{p-1,q-1}. \end{align*}

Equivalently, the cohomological $\partial\bar{\partial}$ lemma identifies the pure-type Bott-Chern representatives with the pure-type pieces of de Rham cohomology.