Dolbeault cohomology on arvau

Wikipedia / 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 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

be the vector bundle. In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections

, 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.

. This satisfies the characteristic Leibniz rule on differential forms, and is therefore sometimes known as a -connection, Therefore, in the same way that a connection on a vector bundle can be extended to the exterior covariant derivative can be extended to an operator

and so Dolbeault cohomology with coefficients in can be defined as above:

Dolbeault–Grothendieck lemma

In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or -Poincaré lemma-Poincaré lemma; we shall use the following generalised form of the Cauchy integral representation for smooth functions:

Proposition, then

Lemma-Poincaré lemma on the complex plane a smooth form, then

Proof. Then we can write

and define

is clearly well-defined and smooth; we note that

we find

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. the open polydisc.

Lemma

-form can be written as

Proof. and observe that we can write

which satisfy

. Define

then

such that

ends the induction step. QED

for some of the components of the polyradius.

Lemma

Proof..

Case 1.; we want to show that

with

be an open set a smooth function such that:

which satisfies

hence the form

satisfies

Case 2. as before, we want to show that

such that

-form wherever defined, hence by the Stone–Weierstrass theorem we can write it as

are polynomials and

but then the form

satisfies

. 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,

is the sheaf of holomorphic p forms on M.

. Namely one has an isomorphism

A version for logarithmic forms has also been established.

Proof

be the fine sheaf-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

-dimensional complex projective space is

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

is a compact Kähler complex manifold and

is the fundamental form associated to the Fubini–Study metric which yields the result.