nLab Dolbeault complex

Idea

The Dolbeault complex is the analog of the de Rham complex in complex geometry.

Definition

Dolbeault complex

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 $q$, hence is given on a local coordinate chart by an expression of the form

$$\omega = \sum f_{I J} d z_{i_1} \wedge \cdots \wedge d z_{i_p} \wedge d \bar z_{j_1} \wedge \cdots \wedge d \bar z_{j_q} \,.$$

Moreover, the de Rham differential $\mathbf{d}$ decomposes as

$$\mathbf{d} = \partial + \bar \partial \,,$$

where $\partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet + 1, \bullet}$ and $\bar \partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet, \bullet + 1}$.

The Dolbeault complex of $X$ is the chain complex $(\Omega^{\bullet, \bullet}(X), \bar \partial)$. The Dolbeault cohomology of $X$ is the cochain cohomology of this complex.

Holomorphic differential forms

Here $\Omega^{p,0}(X)$ defines a holomorphic vector bundle and a holomorphic section is a differential form with local expression as above, such that the coefficient functions $f_{I J}$ are holomorphic functions. This is called a holomorphic differential form.

For $p \lt dim_{\mathbb{C}}(X)$ equivalently this is a differential form in the kernel of the antiholomorphic Dolbeault operator $\bar \partial$.

Properties

Dolbeault theorem

The complex analog of the de Rham theorem is the Dolbeault theorem:

for $X$ a complex manifold then its Dolbeault cohomology in bi-degree $(p,q)$ is naturally isomorphic to the abelian sheaf cohomology in degree $q$ of the abelian sheaf $\Omega^p \coloneqq \Omega^{p,0}$ of holomorphic p-forms

$$H^{p,q}(X)\simeq H^q(X,\Omega^p) \,.$$

(…)

Let $Disk_{compl}$ be the category of complex polydiscs 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 $(p,q)$

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

For instance (Maddock, theorem 1.0.1).

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

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

Related concepts

• Dolbeault cohomology

• Dolbeault-Dirac operator

• chiral Dolbeault complex

• holomorphic de Rham complex

References

• Claire Voisin, section 2.3 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Zachary Maddock, Dolbeault cohomology (pdf)

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