Cohomological Form of the $\partial\bar{\partial}$ Lemma

Theorem

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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.

Discussion

Proof

[proofplan] We first check that the Bott-Chern-to-Aeppli map is meaningful and then prove the central equivalence: injectivity of this map is exactly the statement that a pure-type form which is both $\partial$-closed, $\bar{\partial}$-closed, and Aeppli-exact is already $\partial\bar{\partial}$-exact. This avoids the invalid inference that a single bidegree component of a $d$-exact form is itself $d$-exact. The de Rham representative formulation is then identified through the Bott-Chern-de Rham comparison theorem stated in the formalized statement, which is the compactness-dependent input relating the cohomological $\partial\bar{\partial}$ lemma to the Bott-Chern-to-de Rham comparison maps. [/proofplan] [step:Verify that the comparison map is well-defined] Let $\alpha\in A^{p,q}(X)$ represent a Bott-Chern class. By definition, $\partial\alpha=0$ and $\bar{\partial}\alpha=0$. Therefore \begin{align*} \partial\bar{\partial}\alpha=0, \end{align*} so $\alpha$ also defines an Aeppli class. If $\alpha$ and $\alpha'$ represent the same Bott-Chern class, then there exists $\gamma\in A^{p-1,q-1}(X)$ such that \begin{align*} \alpha'-\alpha=\partial\bar{\partial}\gamma. \end{align*} Since $\partial\bar{\partial}\gamma=\partial(\bar{\partial}\gamma)$ lies in $\partial A^{p-1,q}(X)$, it is zero in Aeppli cohomology. Hence $[\alpha']_A=[\alpha]_A$, so $\iota_{p,q}$ is well-defined. [/step] [step:Show that the cohomological $\partial\bar{\partial}$ lemma implies injectivity of the comparison maps] Assume the cohomological $\partial\bar{\partial}$ lemma. Fix $p,q\ge 0$, and let $[\alpha]_{BC}\in H^{p,q}_{BC}(X)$ satisfy \begin{align*} \iota_{p,q}([\alpha]_{BC})=0. \end{align*} Choose a representative $\alpha\in A^{p,q}(X)$ with $\partial\alpha=0$ and $\bar{\partial}\alpha=0$. The equality $\iota_{p,q}([\alpha]_{BC})=0$ means that $[\alpha]_A=0$ in $H_A^{p,q}(X)$. By the definition of the Aeppli quotient, there exist forms $\mu\in A^{p-1,q}(X)$ and $\nu\in A^{p,q-1}(X)$ such that \begin{align*} \alpha=\partial\mu+\bar{\partial}\nu. \end{align*} Thus $\alpha\in \partial A^{p-1,q}(X)+\bar{\partial}A^{p,q-1}(X)$, while the Bott-Chern representative conditions give $\partial\alpha=0$ and $\bar{\partial}\alpha=0$. The cohomological $\partial\bar{\partial}$ lemma therefore applies directly to $\alpha$ and gives a form $\gamma\in A^{p-1,q-1}(X)$ such that \begin{align*} \alpha=\partial\bar{\partial}\gamma. \end{align*} Hence $[\alpha]_{BC}=0$, so $\iota_{p,q}$ is injective. [/step] [step:Use injectivity to recover the cohomological $\partial\bar{\partial}$ lemma] Assume that $\iota_{p,q}$ is injective for every $p,q\ge 0$. Let $\alpha\in A^{p,q}(X)$ satisfy $\partial\alpha=0$, $\bar{\partial}\alpha=0$, and \begin{align*} \alpha\in \partial A^{p-1,q}(X)+\bar{\partial}A^{p,q-1}(X). \end{align*} The first two equalities say that $\alpha$ represents a Bott-Chern class $[\alpha]_{BC}\in H^{p,q}_{BC}(X)$. The membership condition means that there exist forms $\mu\in A^{p-1,q}(X)$ and $\nu\in A^{p,q-1}(X)$ such that \begin{align*} \alpha=\partial\mu+\bar{\partial}\nu. \end{align*} Thus $[\alpha]_A=0$ in $H_A^{p,q}(X)$. Since $\iota_{p,q}$ is injective and $\iota_{p,q}([\alpha]_{BC})=[\alpha]_A=0$, we obtain $[\alpha]_{BC}=0$. By the definition of Bott-Chern cohomology, there exists $\gamma\in A^{p-1,q-1}(X)$ such that \begin{align*} \alpha=\partial\bar{\partial}\gamma. \end{align*} This is precisely the cohomological $\partial\bar{\partial}$ lemma. [guided] Assume that every comparison map $\iota_{p,q}$ is injective. We want to prove the cohomological $\partial\bar{\partial}$ lemma, so we begin with a pure-type form $\alpha\in A^{p,q}(X)$ satisfying \begin{align*} \partial\alpha=0 \end{align*} and \begin{align*} \bar{\partial}\alpha=0 \end{align*} and \begin{align*} \alpha\in \partial A^{p-1,q}(X)+\bar{\partial}A^{p,q-1}(X). \end{align*} The two closedness conditions are exactly the numerator conditions in Bott-Chern cohomology, so $\alpha$ defines a class $[\alpha]_{BC}\in H^{p,q}_{BC}(X)$. Now use the Aeppli-exactness hypothesis. The statement \begin{align*} \alpha\in \partial A^{p-1,q}(X)+\bar{\partial}A^{p,q-1}(X) \end{align*} means that there are forms $\mu\in A^{p-1,q}(X)$ and $\nu\in A^{p,q-1}(X)$ such that \begin{align*} \alpha=\partial\mu+\bar{\partial}\nu. \end{align*} This equality says exactly that $\alpha$ is zero in Aeppli cohomology, because the denominator of $H_A^{p,q}(X)$ is $\partial A^{p-1,q}(X)+\bar{\partial}A^{p,q-1}(X)$. Therefore \begin{align*} \iota_{p,q}([\alpha]_{BC})=[\alpha]_A=0. \end{align*} By the assumed injectivity of $\iota_{p,q}$, this forces $[\alpha]_{BC}=0$. Finally, the meaning of $[\alpha]_{BC}=0$ is precisely that $\alpha$ lies in the image of $\partial\bar{\partial}$: there exists $\gamma\in A^{p-1,q-1}(X)$ such that \begin{align*} \alpha=\partial\bar{\partial}\gamma. \end{align*} This proves the cohomological $\partial\bar{\partial}$ lemma. [/guided] [/step] [step:Deduce the pure-type de Rham representative statement] Assume the equivalent conditions already proved. We use the Bott-Chern-de Rham comparison theorem in the precise form included in the statement: on the compact complex manifold $X$ of complex dimension $n$ satisfying the cohomological $\partial\bar{\partial}$ lemma, for every integer $k$ with $0\le k\le 2n$ the map $\Phi_k:\bigoplus_{p+q=k}H^{p,q}_{BC}(X)\to H^k_{dR}(X;\mathbb C)$ defined 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*} is an isomorphism. The hypotheses of this comparison theorem match the present situation: $X$ is compact complex by assumption, $0\le k\le 2n$, and the cohomological $\partial\bar{\partial}$ lemma is the equivalent condition established in the preceding steps. The map is well-defined because every Bott-Chern representative is both $\partial$-closed and $\bar{\partial}$-closed, hence $d$-closed by the type decomposition $d=\partial+\bar{\partial}$ for forms on the complex manifold $X$ from [citetheorem:7004]. If $\alpha^{p,q}$ is changed by $\partial\bar{\partial}\gamma^{p-1,q-1}$, then [citetheorem:8046] gives $\bar{\partial}^2=0$, so under the convention $d=\partial+\bar{\partial}$ the total form changes by the $d$-exact form $d(\bar{\partial}\gamma^{p-1,q-1})=\partial\bar{\partial}\gamma^{p-1,q-1}$. Surjectivity of $\Phi_k$ gives, for every de Rham class $[\beta]\in H^k_{dR}(X;\mathbb C)$, Bott-Chern classes $[\beta^{p,q}]_{BC}\in H^{p,q}_{BC}(X)$ such that \begin{align*} [\beta]=\left[\sum_{p+q=k}\beta^{p,q}\right]_{dR}. \end{align*} Choosing Bott-Chern representatives $\beta^{p,q}\in A^{p,q}(X)$ gives $\partial\beta^{p,q}=0$ and $\bar{\partial}\beta^{p,q}=0$, hence $d\beta^{p,q}=0$ by [citetheorem:7004]. Thus \begin{align*} \beta'=\sum_{p+q=k}\beta^{p,q} \end{align*} is a representative of $[\beta]$ whose pure-type components are all $d$-closed. For uniqueness, suppose \begin{align*} \sum_{p+q=k}\beta^{p,q} \end{align*} and \begin{align*} \sum_{p+q=k}\widetilde{\beta}^{p,q} \end{align*} are two such representatives of the same de Rham class. For each $p+q=k$, define \begin{align*} \delta^{p,q}:=\beta^{p,q}-\widetilde{\beta}^{p,q}\in A^{p,q}(X). \end{align*} Since both components are $d$-closed, [citetheorem:7004] implies $\partial\delta^{p,q}=0$ and $\bar{\partial}\delta^{p,q}=0$, so $\delta^{p,q}$ defines a Bott-Chern class. The equality of de Rham classes says \begin{align*} \Phi_k\left(\sum_{p+q=k}[\delta^{p,q}]_{BC}\right)=0. \end{align*} Injectivity of $\Phi_k$ gives $[\delta^{p,q}]_{BC}=0$ for every $p+q=k$. By the definition of Bott-Chern cohomology, 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*} [/step] [step:Recover the cohomological $\partial\bar{\partial}$ lemma from the Bott-Chern-de Rham comparison] Assume the de Rham comparison condition, namely that for every integer $k$ with $0\le k\le 2n$ the map \begin{align*} \Phi_k:\bigoplus_{p+q=k}H^{p,q}_{BC}(X)\longrightarrow H^k_{dR}(X;\mathbb C) \end{align*} is an isomorphism. The converse direction of the Bott-Chern-de Rham comparison theorem stated in the formalized statement applies to the compact complex manifold $X$: if all maps $\Phi_k$ are isomorphisms, then the cohomological $\partial\bar{\partial}$ lemma holds on $X$. Its hypotheses are exactly the compact complex manifold hypothesis and the isomorphism condition just assumed. For completeness, this means that whenever $\alpha\in A^{p,q}(X)$ satisfies $\partial\alpha=0$, $\bar{\partial}\alpha=0$, 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*} By the equivalence already proved between the cohomological $\partial\bar{\partial}$ lemma and injectivity of the maps $\iota_{p,q}$, the comparison maps $\iota_{p,q}$ are injective for every $p,q\ge 0$. Combining the previous steps proves the equivalence of all three formulations. [/step]

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