nLab Dolbeault complex



Complex geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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


Dolbeault complex

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

ω=f IJdz i 1dz i pdz¯ j 1dz¯ j q. \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 d\mathbf{d} decomposes as

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

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

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

Holomorphic differential forms

Here Ω p,0(X)\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 IJf_{I J} are holomorphic functions. This is called a holomorphic differential form.

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


Dolbeault theorem

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

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

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


Let Disk complDisk_{compl} be the category of complex polydiscs 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.


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


(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).


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 \,.


A formal geometry version:

Last revised on October 18, 2023 at 05:59:23. See the history of this page for a list of all contributions to it.