The ∂∂̄-lemma - Knowlpedia

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.

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.