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.
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).
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 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
This satisfies the characteristic Leibniz rule with respect to the Dolbeault operator
, Therefore, in the same way that a connection on a vector bundle can be extended to the exterior covariant derivative, the Dolbeault operator of
The Dolbeault operator satisfies the integrability condition
In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or
-Poincaré lemma; we shall use the following generalised form of the Cauchy integral representation for smooth functions: Proposition: Let
defined above is a well-defined smooth function and
, then we can find a smooth function
applying the generalised Cauchy formula to
Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to Grothendieck.
, hence there exist a family of smooth functions
Define then therefore we can apply the induction hypothesis to it, there exists
QED Lemma (extended Dolbeault-Grothendieck).
, then by the Dolbeault–Grothendieck lemma we can find forms
and we can apply again the Dolbeault-Grothendieck lemma to find a
we cannot apply the Dolbeault-Grothendieck lemma twice; we take
are polynomials and but then the form satisfies which completes the induction step; therefore we have built a sequence
It asserts that the Dolbeault cohomology is isomorphic to the sheaf cohomology of the sheaf of holomorphic differential forms.
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
A version for logarithmic forms has also been established.
The long exact sequences of cohomology corresponding to these give the result, once one uses that the higher cohomologies of a fine sheaf vanish.
-dimensional complex projective space is We apply the following well-known fact from Hodge theory: because
is a compact Kähler complex manifold.
is the fundamental form associated to the Fubini–Study metric (which is indeed Kähler), therefore