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 E is a Dolbeault operator taking sections of E to (0,1)-forms with values in E. This satisfies the characteristic Leibniz rule with respect to the Dolbeault operator on differential forms, and is therefore sometimes known as a (0,1)-connection on E, Therefore, in the same way that a connection on a vector bundle can be extended to the exterior covariant derivative, the Dolbeault operator of E 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 E can be defined as above: The Dolbeault cohomology groups do not depend on the choice of Dolbeault operator compatible with the holomorphic structure of E, 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 0 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 g 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 f_2\equiv 0 in V then g_2 is clearly well-defined and smooth; we note that which is indeed well-defined and smooth, therefore the same is true for g. Now we show that on. since is holomorphic in. applying the generalised Cauchy formula to f_1 we find since, but then on V. Since z 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 q>0 such that, then there exists which satisfies: on Before starting the proof we note that any (p,q)-form can be written as for multi-indices, therefore we can reduce the proof to the case. Proof. Let k>0 be the smallest index such that in the sheaf of -modules, we proceed by induction on k. For k=0 we have since q>0; next we suppose that if then there exists such that on. Then suppose and observe that we can write Since \omega is -closed it follows that \psi, \mu 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 g_J which satisfy g_J 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 Lemma (extended Dolbeault-Grothendieck). If is an open polydisc with and q>0, 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 \beta_i of bidegree (p,q-1) on open such that ; we want to show that We proceed by induction on i: the case when i=1 holds by the previous lemma. Let the claim be true for k>1 and take with Then we find a (p,q-1)-form defined in an open neighbourhood of such that. Let U_k be an open neighbourhood of then on U_k and we can apply again the Dolbeault-Grothendieck lemma to find a (p,q-2)-form \gamma_k such that on \Delta_k. Now, let V_k 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 \beta_i and \Delta_i as before, we want to show that Again, we proceed by induction on i: for i=1 the answer is given by the Dolbeault-Grothendieck lemma. Next we suppose that the claim is true for k>1. We take such that covers, then we can find a (p,0)-form such that which also satisfies on \Delta_k, i.e. is a holomorphic (p,0)-form wherever defined, hence by the Stone–Weierstrass theorem we can write it as where P_I are polynomials and but then the form satisfies which completes the induction step; therefore we have built a sequence which uniformly converges to some (p,0)-form \beta such that. 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 \Omega^p 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 E. Namely one has an isomorphism A version for logarithmic forms has also been established.

Proof

Let be the fine sheaf of C^{\infty} forms of type (p,q). 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 calculation

The Dolbeault cohomology of the n-dimensional complex projective space is We apply the following well-known fact from Hodge theory: because is a compact Kähler complex manifold. Then b_{2k+1}=0 and Furthermore we know that is Kähler, and where \omega is the fundamental form associated to the Fubini–Study metric (which is indeed Kähler), therefore h^{k,k}=1 and h^{p,q}=0 whenever p\ne q, which yields the result.

Footnotes