Dolbeault cohomology - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold. Its Dolbeault cohomology group of bidegree $(p,q)$ is the cohomology

H_{\bar\partial}^{p,q}(X) =\frac{\ker\bigl(\bar\partial:\Omega^{p,q}(X)\to\Omega^{p,q+1}(X)\bigr)} {\operatorname{im}\bigl(\bar\partial:\Omega^{p,q-1}(X)\to\Omega^{p,q}(X)\bigr)}

of the Dolbeault complex in degree $q$ . Its elements are classes of smooth $\bar\partial$ -closed $(p,q)$ -forms, with two representatives equivalent when their difference is $\bar\partial$ -exact. This is a complex vector space depending only on the complex structure of $X$ , not on a chosen Hermitian metric. Both bidegrees are part of the invariant and must be specified.

Sheaf-cohomological interpretation

The Dolbeault theorem gives a natural isomorphism

H_{\bar\partial}^{p,q}(X)\cong H^q(X,\Omega_X^p),

where $\Omega_X^p$ is the sheaf of holomorphic $p$ -forms. The proof uses the Dolbeault lemma and the fine resolution by smooth $(p,\bullet)$ -forms Wells, Chapter II, §3, Theorem 3.17. This identifies an analytic quotient of differential forms with a sheaf-cohomological invariant.

Products and functoriality

Wedge product descends to a bigraded product

H_{\bar\partial}^{p,q}(X)\times H_{\bar\partial}^{r,s}(X) \longrightarrow H_{\bar\partial}^{p+r,q+s}(X).

A holomorphic map $f:X\to Y$ preserves form type and commutes with $\bar\partial$ , so pullback induces maps $f^*:H_{\bar\partial}^{p,q}(Y)\to H_{\bar\partial}^{p,q}(X)$ .

In degree $q=0$ , there are no incoming coboundaries, and $H_{\bar\partial}^{p,0}(X)$ is exactly the space of global holomorphic $p$ -forms. A $\bar\partial$ -closed form that is itself $\bar\partial$ -exact represents zero rather than a new class.

Relation to de Rham cohomology

Dolbeault cohomology keeps track of bidegree, whereas de Rham cohomology uses the total exterior derivative. On a compact Kähler manifold, Hodge theory yields the decomposition

H^k_{\mathrm{dR}}(X;\mathbb C) \cong\bigoplus_{p+q=k}H_{\bar\partial}^{p,q}(X),

but such a direct-sum identification need not hold for a general complex manifold Voisin, §6.1.

References

R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: Chapter II, §3, especially Theorem 3.17, the Dolbeault theorem.
Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. DOI record. Relevant: §2.3.3 for the Dolbeault complex and §6.1 for Hodge decomposition.