Dolbeault cohomology

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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 Hp,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).

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

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

Since

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

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

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

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 $\mathcal O(E)$ of holomorphic sections of E. This is therefore a recollection of the sheaf cohomology of $\mathcal O(E)$.

Dolbeault's theorem

Dolbeault's theorem is a complex analog[1] 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 Ωp is the sheaf of holomorphic p forms on M.

Proof

Let $\mathcal{F}^{p,q}$ be the fine sheaf of $C^{\infty}$ forms of type $(p,q)$. Then the $\overline{\partial}$-Poincaré lemma says that the sequence

$\Omega^{p,q} \xrightarrow{\overline{\partial}} \mathcal{F}^{p,q+1}\xrightarrow{\overline{\partial}} \mathcal{F}^{p,q+2} \xrightarrow{\overline{\partial}} \cdots \,$

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.

Footnotes

1. ^ In constrast to De Rham cohomology, Dolbeault cohomology is no longer a topological invariant because it depends closely on complex structure.

References

• Dolbeault, P. (1953). "Sur la cohomologie des variétés analytiques complexes". C. R. Acad. Sci. Paris 236: 175–277.
• Wells, R.O. (1980). Differential Analysis on Complex Manifolds. Springer-Verlag. ISBN 0-387-90419-0.