nLab Dolbeault complex

complex geometry

Examples

Differential geometry

differential geometry

synthetic differential geometry

Contents

Idea

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

(…)

Properties

In terms of sheaf cohomology

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

For $p \in \mathbb{N}$ write $\Omega^p \colon Disk_{complex}^{op} \to Set$ 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 $(p,q)$

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

On Stein manifolds

Proposition

(Cartan theorem B)

For $X$ a Stein manifold,

$H^k(\Omega^{p,\bullet}(X), \bar \partial) = \left\{ \array{ 0 & k \neq 0 \\ \Omega^p_{hol}(X) & k = 0 } \right. \,.$

For instance (Gunning-Rossi).

Proposition

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

$H^k(\Omega_c^{p, \bullet}(X), \bar \partial) = \left\{ \array{ 0 , & k \neq n \\ (\Omega_{hol}^{n-p}(X))^\ast } \right. \,,$

where on the right $(-)^\ast$ denotes the continuous linear dual.

First noticed in (Serre).

Todd genus

By the Hirzebruch-Riemann-Roch theorem the index of the Dolbeault operator is the Todd genus.

Relation to $Spin^c$-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

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

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 March 21, 2014 09:09:15 by Urs Schreiber (89.204.138.115)