In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).
Construction of the cohomology groups
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
Dolbeault cohomology of vector bundles
If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution of the sheaf of holomorphic sections of E, using the Dolbeault operator of E. This is therefore a resolution of the sheaf cohomology of .
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
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
Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to Grothendieck. 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