nLab
Dolbeault complex

Idea

On a complex manifold $X$ the de Rham complex ${\Omega }^{•}(X)$ refines to a bigraded complex ${\Omega }^{•,•}(X)$, where a differential form of bidegree $(p,q)$ has holomorphic degree $p$ and antiholomorphic degree $q$.

Definition

(…)

Properties

In terms of sheaf cohomology

Let ${\mathrm{Disk}}_{\mathrm{compl}}$ be the category of complex unit disks in ${ℂ}^n$ and holomorphic functions between them.

For $p\in \mathbb{N}$ write ${\Omega }^p:{\mathrm{Disk}}_{\mathrm{complex}}^{\mathrm{op}}\to \mathrm{Set}$ for the sheaf of holomorphic differential p-forms.

Proposition

For $X$ a complex manifold, let $\{U_i\to X\}$ be a holomorphic good open cover. Then the Cech cohomology of this cover with coefficients in ${\Omega }^p$ in degree $q$ is the Dolbeault cohomology in bidegree $(p,q)$

$$H^{p,q}(X)\simeq {\pi }_0\mathrm{sPSh}({\mathrm{Disk}}_{\mathrm{comp}})(C(\{U_i\},{\Omega }^p[q])).$$H^{p,q}(X) \simeq \pi_0 sPSh(Disk_{comp})(C(\{U_i\}, \Omega^p[q])) \,.

For instance (Maddock, theorem 1.0.1).

On Stein manifolds

Proposition

(Cartan theorem B)

For $X$ a Stein manifold,

$$H^k({\Omega }^{p,•}(X),\overline{\partial })=\{\begin{array}{cc}0& k\ne 0\\ {\Omega }_{\mathrm{hol}}^p(X)& k=0\end{array}.$$H^k(\Omega^{p,\bullet}(X), \bar \partial) = \left\{ \array{ 0 & k \neq 0 \\ \Omega^p_{hol}(X) & k = 0 } \right. \,.

For instance (Gunning-Rossi).

Proposition

For $X$ a Stein manifold of complex dimension $n$, the compactly supported Dolbeault cohomology is

$$H^k({\Omega }_c^{p,•}(X),\overline{\partial })=\{\begin{array}{cc}0,& k\ne n\\ ({\Omega }_{\mathrm{hol}}^{n-p}(X){)}^{*}\end{array},$$H^k(\Omega_c^{p, \bullet}(X), \bar \partial) = \left\{ \array{ 0 , & k \neq n \\ (\Omega_{hol}^{n-p}(X))^\ast } \right. \,,

where on the right $(-{)}^{*}$ denotes the continuous linear dual.

First noticed in (Serre).

Relation to ${\mathrm{Spin}}^c$-structures

A complex manifold, being in particular an almost complex manifold, carries a canonical spin^c structure. The corresponding Spin^c Dirac operator identifies with the Dolbeault operator under the identification of the spinor bundle with that of holomorphic differential forms

$$S(X)\simeq {\wedge }^{0,•}{T}^{*}X.$$S(X) \simeq \wedge^{0,\bullet} T^\ast X \,.

References

• Zachary Maddock, Dolbeault cohomology (pdf)

• Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs, N.J., (1965)

• Jean-Pierre Serre, Quelques problèmes globaux relatifs aux variétés de Stein, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège, 1953, pp. 57–68. MR 0064155 (16,235b)

A formal geometry version:

• Shilin Yu, Dolbeault dga of a formal neighborhood, arxiv/1206.5155; The Dolbeault dga of the formal neighborhood of a diagonal, arxiv/1211.1567