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