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Dolbeault complex

Context

Complex geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

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

Definition

(…)

Properties

In terms of sheaf cohomology

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

For pp \in \mathbb{N} write Ω p:Disk complex opSet\Omega^p \colon Disk_{complex}^{op} \to Set for the sheaf of holomorphic differential p-forms.

Proposition

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

H p,q(X)π 0sPSh(Disk comp)(C({U i},Ω 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 XX a Stein manifold,

H k(Ω p,(X),¯)={0 k0 Ω hol p(X) k=0. 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 XX a Stein manifold of complex dimension nn, the compactly supported Dolbeault cohomology is

H k(Ω c p,(X),¯)={0, kn (Ω hol np(X)) *, 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 cSpin^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) 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:

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