ddbar lemma

is a complex differential form of bidegree (p,q) (with

is zero in de Rham cohomology, then there exists a form

are the Dolbeault operators of the complex manifold

is a differential form with real coefficients, then so is

This lemma should be compared to the notion of an exact differential form in de Rham cohomology.

is a closed differential k-form (on any smooth manifold) whose class is zero in de Rham cohomology, then

Indeed, since the Dolbeault operators sum to give the exterior derivative

-potential in the setting of compact Kähler manifolds.

-lemma is a consequence of Hodge theory applied to a compact Kähler manifold.

[3][1]: 41–44 [2]: 73–77 The Hodge theorem for an elliptic complex may be applied to any of the operators

To these operators one can define spaces of harmonic differential forms given by the kernels:

3.2.8  These decompositions hold separately on any compact complex manifold.

The importance of the manifold being Kähler is that there is a relationship between the Laplacians of

be a closed (p,q)-form on a compact Kähler manifold

then since this sum is from an orthogonal decomposition with respect to the inner product

-lemma holds and can be proven without the need to appeal to the Hodge decomposition theorem.

Then by the Poincaré lemma there exists an open neighbourhood

After possibly shrinking the size of the open neighbourhood

In particular when a compact complex manifold is a Kähler manifold, the Bott–Chern cohomology is isomorphic to the Dolbeault cohomology, but in general it contains more information.

The Bott–Chern cohomology groups of a compact complex manifold[3] are defined by

In this way, the kernel of the maps above measure the failure of the manifold

-lemma occurs when the complex differential form has bidegree (1,1).

In this case the lemma states that an exact differential form

is a Kähler form restricted to a small open subset

of a Kähler manifold (this case follows from the local version of the lemma), where the aforementioned Poincaré lemma ensures that it is an exact differential form.

This leads to the notion of a Kähler potential, a locally defined function which completely specifies the Kähler form.

is the difference of two Kähler forms which are in the same de Rham cohomology class

By allowing (differences of) Kähler forms to be completely described using a single function, which is automatically a plurisubharmonic function, the study of compact Kähler manifolds can be undertaken using techniques of pluripotential theory, for which many analytical tools are available.

Complex manifolds which are not necessarily Kähler but still happen to satisfy the

For example, compact complex manifolds which are Fujiki class C satisfy the