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 formula_1 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 Ω"p","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 formula_5 of holomorphic sections of "E", using the Dolbeault operator of "E". This is therefore a resolution of the sheaf cohomology of formula_5.
Dolbeault–Grothendieck lemma.
In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or formula_7-Poincaré lemma). First we prove a one-dimensional version of the formula_7-Poincaré lemma; we shall use the following generalised form of the Cauchy integral representation for smooth functions:
Proposition: Let formula_9 the open ball centered in formula_10 of radius formula_11 formula_12 open and formula_13, then 
Lemma (formula_7-Poincaré lemma on the complex plane): Let formula_16 be as before and formula_17 a smooth form, then
satisfies formula_19 on formula_20
"Proof." Our claim is that formula_21 defined above is a well-defined smooth function such that formula_22 is locally formula_7-exact. To show this we choose a point formula_24 and an open neighbourhood formula_25, then we can find a smooth function formula_26 whose support is compact and lies in formula_27 and formula_28 Then we can write 
and define
Since formula_31 in formula_32 then formula_33 is clearly well-defined and smooth; we note that
which is indeed well-defined and smooth, therefore the same is true for formula_21. Now we show that formula_36 on formula_27.
since formula_39 is holomorphic in formula_40 .
applying the generalised Cauchy formula to formula_42 we find
since formula_44, but then formula_45 on formula_46. QED
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 formula_47 the open polydisc centered in formula_48 with radius formula_49.
Lemma (Dolbeault–Grothendieck): Let formula_50 where formula_51 open and formula_52 such that formula_53, then there exists formula_54 which satisfies: formula_55 on formula_56
Before starting the proof we note that any formula_57-form can be written as
for multi-indices formula_59, therefore we can reduce the proof to the case formula_60.
"Proof." Let formula_61 be the smallest index such that formula_62 in the sheaf of formula_63-modules, we proceed by induction on formula_64. For formula_65 we have formula_66 since formula_52; next we suppose that if formula_62 then there exists formula_69 such that formula_55 on formula_47. Then suppose formula_72 and observe that we can write
Since formula_74 is formula_7-closed it follows that formula_76 are holomorphic in variables formula_77 and smooth in the remaining ones on the polydisc formula_47. Moreover we can apply the formula_7-Poincaré lemma to the smooth functions formula_80 on the open ball formula_81, hence there exist a family of smooth functions formula_82 which satisfy 
formula_82 are also holomorphic in formula_77. Define 
then
therefore we can apply the induction hypothesis to it, there exists formula_88 such that
and formula_90 ends the induction step. QED
Lemma (extended Dolbeault-Grothendieck). If formula_92 is an open polydisc with formula_93 and formula_52, then formula_95
"Proof." We consider two cases: formula_96 and formula_97.
Case 1. Let formula_96, and we cover formula_92 with polydiscs formula_100, then by the Dolbeault–Grothendieck lemma we can find forms formula_101 of bidegree formula_102 on formula_103 open such that formula_104; we want to show that 
We proceed by induction on formula_106: the case when formula_107 holds by the previous lemma. Let the claim be true for formula_108 and take formula_109 with 
Then we find a formula_102-form formula_112 defined in an open neighbourhood of formula_113 such that formula_114. Let formula_115 be an open neighbourhood of formula_116 then formula_117 on formula_115 and we can apply again the Dolbeault-Grothendieck lemma to find a formula_119-form formula_120 such that formula_121 on formula_122. Now, let formula_123 be an open set with formula_124 and formula_125 a smooth function such that: 
Then formula_127 is a well-defined smooth form on formula_92 which satisfies 
hence the form 
satisfies
Case 2. If instead formula_132 we cannot apply the Dolbeault-Grothendieck lemma twice; we take formula_101 and formula_134 as before, we want to show that 
Again, we proceed by induction on formula_136: for formula_137 the answer is given by the Dolbeault-Grothendieck lemma. Next we suppose that the claim is true for formula_138. We take formula_139 such that formula_140 covers formula_92, then we can find a formula_142-form formula_112 such that 
which also satisfies formula_117 on formula_122, i.e. formula_147 is a holomorphic formula_142-form wherever defined, hence by the Stone–Weierstrass theorem we can write it as
where formula_150 are polynomials and 
but then the form 
satisfies
which completes the induction step; therefore we have built a sequence formula_154 which uniformly converges to some formula_142-form formula_156 such that formula_157. QED
Dolbeault's theorem.
Dolbeault's theorem is a complex analog 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 formula_159 is the sheaf of holomorphic "p" forms on "M".
A version for logarithmic forms has also been established.
Proof.
Let formula_160 be the fine sheaf of formula_161 forms of type formula_57. Then the formula_163-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 calculation.
The Dolbeault cohomology of the formula_165-dimensional complex projective space is
We apply the following well-known fact from Hodge theory: 
because formula_168 is a compact Kähler complex manifold. Then formula_169 and 
Furthermore we know that formula_168 is Kähler, and formula_172 where formula_74 is the fundamental form associated to the Fubini–Study metric (which is indeed Kähler), therefore formula_174 and formula_175 whenever formula_176 which yields the result.