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Concept
Dolbeault cohomology
Summary
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).
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 .
In particular associated to the holomorphic structure of is a Dolbeault operator taking sections of to -forms with values in . This satisfies the characteristic Leibniz rule with respect to the Dolbeault operator on differential forms, and is therefore sometimes known as a -connection on , Therefore, in the same way that a connection on a vector bundle can be extended to the exterior covariant derivative, the Dolbeault operator of 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' can be defined as above:
The Dolbeault cohomology groups do not depend on the choice of Dolbeault operator compatible with the holomorphic structure of , so are typically denoted by dropping the dependence on .
In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or -Poincaré lemma).
Official source
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