Let Ωp,q be the vector bundle of complex differential forms of degree (p,q). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections
Since
this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space
In particular associated to the holomorphic structure of is a Dolbeault operator taking sections of to -forms with values in . This satisfies the characteristic Leibniz rule with respect to the Dolbeault operator on differential forms, and is therefore sometimes known as a -connection on , Therefore, in the same way that a connection on a vector bundle can be extended to the exterior covariant derivative, the Dolbeault operator of can be extended to an operator
which acts on a section by
and is extended linearly to any section in . The Dolbeault operator satisfies the integrability condition and so Dolbeault cohomology with coefficients in can be defined as above:
The Dolbeault cohomology groups do not depend on the choice of Dolbeault operator compatible with the holomorphic structure of , so are typically denoted by dropping the dependence on .
Dolbeault–Grothendieck lemma
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In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or -Poincaré lemma). First we prove a one-dimensional version of the -Poincaré lemma; we shall use the following generalised form of the Cauchy integral representation for smooth functions:
Proposition: Let the open ball centered in of radius open and , then
Lemma (-Poincaré lemma on the complex plane): Let be as before and a smooth form, then
satisfies on
Proof. Our claim is that defined above is a well-defined smooth function and . To show this we choose a point and an open neighbourhood , then we can find a smooth function whose support is compact and lies in and Then we can write
and define
Since in then is clearly well-defined and smooth; we note that
which is indeed well-defined and smooth, therefore the same is true for . Now we show that on .
since is holomorphic in .
applying the generalised Cauchy formula to we find
since , but then on . Since was arbitrary, the lemma is now proved.
Proof of Dolbeault–Grothendieck lemma
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Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to Grothendieck.[1][2] We denote with the open polydisc centered in with radius .
Lemma (Dolbeault–Grothendieck): Let where open and such that , then there exists which satisfies: on
Before starting the proof we note that any -form can be written as
for multi-indices , therefore we can reduce the proof to the case .
Proof. Let be the smallest index such that in the sheaf of -modules, we proceed by induction on . For we have since ; next we suppose that if then there exists such that on . Then suppose and observe that we can write
Since is -closed it follows that are holomorphic in variables and smooth in the remaining ones on the polydisc . Moreover we can apply the -Poincaré lemma to the smooth functions on the open ball , hence there exist a family of smooth functions which satisfy
are also holomorphic in . Define
then
therefore we can apply the induction hypothesis to it, there exists such that
and ends the induction step. QED
The previous lemma can be generalised by admitting polydiscs with for some of the components of the polyradius.
Lemma (extended Dolbeault-Grothendieck). If is an open polydisc with and , then
Proof. We consider two cases: and .
Case 1. Let , and we cover with polydiscs , then by the Dolbeault–Grothendieck lemma we can find forms of bidegree on open such that ; we want to show that
We proceed by induction on : the case when holds by the previous lemma. Let the claim be true for and take with
Then we find a -form defined in an open neighbourhood of such that . Let be an open neighbourhood of then on and we can apply again the Dolbeault-Grothendieck lemma to find a -form such that on . Now, let be an open set with and a smooth function such that:
Then is a well-defined smooth form on which satisfies
hence the form
satisfies
Case 2. If instead we cannot apply the Dolbeault-Grothendieck lemma twice; we take and as before, we want to show that
Again, we proceed by induction on : for the answer is given by the Dolbeault-Grothendieck lemma. Next we suppose that the claim is true for . We take such that covers , then we can find a -form such that
which also satisfies on , i.e. is a holomorphic -form wherever defined, hence by the Stone–Weierstrass theorem we can write it as
where are polynomials and
but then the form
satisfies
which completes the induction step; therefore we have built a sequence which uniformly converges to some -form such that . QED
Dolbeault's theorem
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Dolbeault's theorem is a complex analog[3] of de Rham's theorem. It asserts that the Dolbeault cohomology is isomorphic to the sheaf cohomology of the sheaf of holomorphic differential forms. Specifically,
where is the sheaf of holomorphic p forms on M.
A version of the Dolbeault theorem also holds for Dolbeault cohomology with coefficients in a holomorphic vector bundle . Namely one has an isomorphism
Let be the fine sheaf of forms of type . Then the -Poincaré lemma says that the sequence
is exact. Like any long exact sequence, this sequence breaks up into short exact sequences. The long exact sequences of cohomology corresponding to these give the result, once one uses that the higher cohomologies of a fine sheaf vanish.
Furthermore we know that is Kähler, and where is the fundamental form associated to the Fubini–Study metric (which is indeed Kähler), therefore and whenever which yields the result.
-lemma, which describes the potential of a -exact differential form in the setting of compact Kähler manifolds.
Footnotes
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^Serre, Jean-Pierre (1953–1954), "Faisceaux analytiques sur l'espace projectif", Séminaire Henri Cartan, 6 (Talk no. 18): 1–10
^"Calculus on Complex Manifolds". Several Complex Variables and Complex Manifolds II. 1982. pp. 1–64. doi:10.1017/CBO9780511629327.002. ISBN 9780521288880.
^In contrast to de Rham cohomology, Dolbeault cohomology is no longer a topological invariant because it depends closely on complex structure.
^Navarro Aznar, Vicente (1987), "Sur la théorie de Hodge–Deligne", Inventiones Mathematicae, 90 (1): 11–76, Bibcode:1987InMat..90...11A, doi:10.1007/bf01389031, S2CID 122772976, Section 8