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 .[1]: 1.17 [2]: Lem 5.50 

Statement

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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  .[3]: Ch VI Lem 8.6 

ddbar potential

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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

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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:

  1.   is  -exact.
  2.   is  -exact.
  3.   is  -exact.
  4.   is  -exact. That is there exists   such that  .
  5.   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.

Local version

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A local version of the  -lemma holds and can be proven without the need to appeal to the Hodge decomposition theorem.[4]: Ex 1.3.3, Rmk 3.2.11  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.

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 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  .

Bott–Chern cohomology

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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)

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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  :

 

In particular this occurs in the case where   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. Another important case is when   is the difference of two Kähler forms which are in the same de Rham cohomology class  . In this case   in de Rham cohomology so the  -lemma applies. 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. For example, the  -lemma is used to rephrase the Kähler–Einstein equation in terms of potentials, transforming it into a complex Monge–Ampère equation for the Kähler potential.

ddbar manifolds

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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]

See also

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References

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  1. ^ a b Gauduchon, P. (2010). "Elements of Kähler geometry". Calabi's extremal Kähler metrics: An elementary introduction (Preprint).
  2. ^ a b Ballmann, Werner (2006). Lectures on Kähler Manifolds. European mathematical society. doi:10.4171/025. ISBN 978-3-03719-025-8.
  3. ^ a b c d Demailly, Jean-Pierre (2012). Analytic Methods in Algebraic Geometry. Somerville, MA: International Press. ISBN 9781571462343.
  4. ^ a b c d Huybrechts, D. (2005). Complex Geometry. Universitext. Berlin: Springer. doi:10.1007/b137952. ISBN 3-540-21290-6.
  5. ^ a b Angella, Daniele; Tomassini, Adriano (2013). "On the  -Lemma and Bott-Chern cohomology". Inventiones Mathematicae. 192: 71–81. arXiv:1402.1954. doi:10.1007/s00222-012-0406-3. S2CID 253747048.
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