The ∂∂̄-lemma - Imported Knowlpedia Content

Definition

Let $X$ be a compact Kähler manifold, and let $\alpha$ be a complex-valued form of pure type $(p,q)$ with $d\alpha=0$ . The $\partial\bar\partial$ -lemma states that the following are equivalent: $\alpha$ is $d$ -exact, $\partial$ -exact, $\bar\partial$ -exact, or $\partial\bar\partial$ -exact. Thus, whenever any of the first three conditions holds, there is a $(p-1,q-1)$ -form $\beta$ such that

\alpha=\partial\bar\partial\beta.

The compactness and Kähler hypotheses are essential parts of this statement; the lemma can fail on general complex manifolds.

Equivalent formulations

In terms of images and kernels on all complex-valued forms, one common formulation is

\ker\partial\cap\ker\bar\partial\cap (\operatorname{im}\partial+\operatorname{im}\bar\partial) =\operatorname{im}(\partial\bar\partial).

With the convention $d^c=i(\bar\partial-\partial)$ from the $d^c$ -operator, this is equivalently expressed through

\operatorname{im}d\cap\ker d^c =\operatorname{im}(dd^c) =\ker d\cap\operatorname{im}d^c.

These formulations differ only by the nonzero scalar relating $dd^c$ and $\partial\bar\partial$ .

Structure and consequences

The lemma identifies several apparently different notions of triviality. In particular, Bott–Chern cohomology maps isomorphically to Dolbeault cohomology in each bidegree, while the direct sum of its $(p,q)$ -groups with $p+q=k$ maps isomorphically to degree- $k$ de Rham cohomology. It also ensures that this Hodge decomposition is independent of the chosen Kähler metric. Demailly proves these consequences in Chapter VI, §8.2, Lemma 8.6 and Corollary 8.7.

Conventions and scope

The pure-type formulation requires $d\alpha=0$ . Merely assuming $\bar\partial\alpha=0$ does not make $d$ -, $\partial$ -, and $\bar\partial$ -exactness equivalent. Compact non-Kähler complex manifolds provide genuine failures, so this lemma must not be applied from complex structure alone.

References

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter VI, §8.2, especially Lemma 8.6 and Corollary 8.7.