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
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.[3][1]: 41–44 [2]: 73–77
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.[4]: Thm. 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 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.[4]: Prop. 3.1.12
As a result of the above decompositions, one can prove the following lemma.
Lemma (-lemma)[3]: 311 — Let be a -closed (p,q)-form on a compact Kähler manifold . Then the following are equivalent:
is -exact.
is -exact.
is -exact.
is -exact. That is there exists such that .
is orthogonal to .
The proof is as follows.[4]: Cor. 3.2.10 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.
Lemma (Local -lemma) — Let be a complex manifold and be a differential form of bidegree (p,q) for . Then is -closed if and only if for every point there exists an open neighbourhood containing and a differential form such that on .
The Bott–Chern cohomology is a cohomology theory for compact complex manifolds which depends on the operators and , and measures the extent to which the -lemma fails to hold. 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
Since a differential form which is both and -closed is -closed, there is a natural map from Bott–Chern cohomology groups to de Rham cohomology groups. There are also maps to the and Dolbeault cohomology groups . When the manifold satisfies the -lemma, for example if it is a compact Kähler manifold, then the above maps from Bott–Chern cohomology to Dolbeault cohomology are isomorphisms, and furthermore the map from Bott–Chern cohomology to de Rham cohomology is injective.[5] As a consequence, there is an isomorphism
whenever satisfies the -lemma. In this way, the kernel of the maps above measure the failure of the manifold to satisfy the lemma, and in particular measure the failure of to be a Kähler manifold.
Consequences for bidegree (1,1)
The most significant consequence of the -lemma occurs when the complex differential form has bidegree (1,1). In this case the lemma states that an
exact differential form has a -potential given by a smooth function :
Complex manifolds which are not necessarily Kähler but still happen to satisfy the -lemma are known as -manifolds. For example, compact complex manifolds which are
Fujiki class C satisfy the -lemma but are not necessarily Kähler.[5]