Dolbeault cohomology

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 groupsEdit

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 bundlesEdit

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  .

Dolbeault–Grothendieck lemmaEdit

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 such that   is locally  -exact. 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  . QED

Proof of Dolbeault–Grothendieck lemmaEdit

Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to Grothendieck.[1] 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 theoremEdit

Dolbeault's theorem is a complex analog[2] 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 for logarithmic forms has also been established.[3]

ProofEdit

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.

Explicit example of calculationEdit

The Dolbeault cohomology of the  -dimensional complex projective space is

 

We apply the following well-known fact from Hodge theory:

 

because   is a compact Kähler complex manifold. Then   and

 

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.

See alsoEdit

FootnotesEdit

  1. ^ Serre, Jean-Pierre (1953–1954), "Faisceaux analytiques sur l'espace projectif", Séminaire Henri Cartan, 6 (Talk no. 18): 1–10
  2. ^ In contrast to de Rham cohomology, Dolbeault cohomology is no longer a topological invariant because it depends closely on complex structure.
  3. ^ Navarro Aznar, Vicente (1987), "Sur la théorie de Hodge–Deligne", Inventiones Mathematicae, 90 (1): 11–76, doi:10.1007/bf01389031, Section 8

ReferencesEdit