In complex geometry, the lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The -lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold. Sometimes it is also known as the -lemma, due to the use of a related operator , with the relation between the two operators being and so .
Statement
The lemma asserts that if is a compact Kähler manifold and is a complex differential form of bidegree (p,q) (with ) whose class is zero in de Rham cohomology, then there exists a form of bidegree (p-1,q-1) such that
where and are the Dolbeault operators of the complex manifold .
ddbar potential
The form is called the -potential of . The inclusion of the factor ensures that is a real differential operator, that is if 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. In particular if is a closed differential k-form (on any smooth manifold) whose class is zero in de Rham cohomology, then for some differential (k-1)-form called the -potential (or just potential) of , where is the exterior derivative. Indeed, since the Dolbeault operators sum to give the exterior derivative and square to give zero , the -lemma implies that , refining the -potential to the -potential in the setting of compact Kähler manifolds.
Proof
The -lemma is a consequence of Hodge theory applied to a compact Kähler manifold.
The Hodge theorem for an elliptic complex may be applied to any of the operators and respectively to their Laplace operators . To these operators one can define spaces of harmonic differential forms given by the kernels:
The Hodge decomposition theorem asserts that there are three orthogonal decompositions associated to these spaces of harmonic forms, given by
where are the formal adjoints of with respect to the Riemannian metric of the Kähler manifold, respectively. 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 and hence of the orthogonal decompositions above. In particular on a compact Kähler manifold
which implies an orthogonal decomposition
where there are the further relations relating the spaces of and -harmonic forms.
As a result of the above decompositions, one can prove the following lemma.
The proof is as follows. Let be a closed (p,q)-form on a compact Kähler manifold . It follows quickly that (d) implies (a), (b), and (c). Moreover, the orthogonal decompositions above imply that any of (a), (b), or (c) imply (e). Therefore, the main difficulty is to show that (e) implies (d).
To that end, suppose that is orthogonal to the subspace . Then . Since is -closed and , it is also -closed (that is ). If where and is contained in then since this sum is from an orthogonal decomposition with respect to the inner product induced by the Riemannian metric,
or in other words and . Thus it is the case that . This allows us to write for some differential form . Applying the Hodge decomposition for to ,
where is -harmonic, and . The equality implies that is also -harmonic and therefore . Thus . However, since is -closed, it is also -closed. Then using a similar trick to above,
also applying the Kähler identity that . Thus and setting produces the -potential.
Local version
A local version of the -lemma holds and can be proven without the need to appeal to the Hodge decomposition theorem. It is the analogue of the Poincaré lemma or Dolbeault–Grothendieck lemma for the operator. The local -lemma holds over any domain on which the aforementioned lemmas hold.
The proof follows quickly from the aforementioned lemmas. Firstly observe that if is locally of the form for some then because , , and . On the other hand, suppose is -closed. Then by the Poincaré lemma there exists an open neighbourhood of any point and a form such that . Now writing for and note that and comparing the bidegrees of the forms in implies that and and that . After possibly shrinking the size of the open neighbourhood , the Dolbeault–Grothendieck lemma may be applied to and (the latter because ) to obtain local forms such that and . Noting then that this completes the proof as where .
