# nLab Dolbeault complex

complex geometry

### Examples

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

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

(…)

## 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 $p\in ℕ$ 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 $\left\{{U}_{i}\to X\right\}$ 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 $\left(p,q\right)$

${H}^{p,q}\left(X\right)\simeq {\pi }_{0}\mathrm{sPSh}\left({\mathrm{Disk}}_{\mathrm{comp}}\right)\left(C\left(\left\{{U}_{i}\right\},{\Omega }^{p}\left[q\right]\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$H^{p,q}(X) \simeq \pi_0 sPSh(Disk_{comp})(C(\{U_i\}, \Omega^p[q])) \,.

### On Stein manifolds

###### Proposition

(Cartan theorem B)

For $X$ a Stein manifold,

${H}^{k}\left({\Omega }^{p,•}\left(X\right),\overline{\partial }\right)=\left\{\begin{array}{cc}0& k\ne 0\\ {\Omega }_{\mathrm{hol}}^{p}\left(X\right)& k=0\end{array}\phantom{\rule{thinmathspace}{0ex}}.$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

${H}^{k}\left({\Omega }_{c}^{p,•}\left(X\right),\overline{\partial }\right)=\left\{\begin{array}{cc}0,& k\ne n\\ \left({\Omega }_{\mathrm{hol}}^{n-p}\left(X\right){\right)}^{*}\end{array}\phantom{\rule{thinmathspace}{0ex}},$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 $\left(-{\right)}^{*}$ 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

$S\left(X\right)\simeq {\wedge }^{0,•}{T}^{*}X\phantom{\rule{thinmathspace}{0ex}}.$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:

Revised on July 12, 2013 17:37:06 by Urs Schreiber (82.113.121.27)