Dolbeault cohomology

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In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is a special generalization of de Rham cohomology to complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups H^p,q(M,C) 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).

[edit] 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

$\bar{\partial}:\Gamma(\Omega^{p,q})\rightarrow\Gamma(\Omega^{p,q+1})$

Since

$\bar{\partial}^2=0$

so this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space

$H^{p,q}(M,\mathbb{C})=\frac{\hbox{ker}\left(\bar{\partial}:\Gamma(\Omega^{p,q},M)\rightarrow \Gamma(\Omega^{p,q+1},M)\right)}{\bar{\partial}\Gamma(\Omega^{p,q-1})}.$

[edit] 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,

$H^{p,q}(M)\cong H^q(M,\Omega^p)$

where Ω^p is the sheaf of holomorphic p forms on M.

[edit] Proof

The proof is essentially the same as for the sheaf-theoretic version of de Rham's theorem. The primary difference is that instead of the Poincaré lemma, one must employ a version of the lemma adapted to the Dolbeault operator. The rest of the proof carries through after making the necessary adjustments.

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Categories: Homology theory | Differential forms | Topological methods of algebraic geometry | Complex manifolds