geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The Dolbeaul complex is the analog of the de Rham complex in complex geometry.
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
Moreover, the de Rham differential $\mathbf{d}$ decomposes as
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.
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 Dolbeaul operator $\bar \partial$.
The complex analog of the de Rham theorem is the Dolbeault theorem:
for $X$ a complex manifold then ints 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
(…)
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.
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 $(p,q)$
For instance (Maddock, theorem 1.0.1).
For $X$ a Stein manifold,
For instance (Gunning-Rossi).
For $X$ a Stein manifold of complex dimension $n$, the compactly supported Dolbeault cohomology is
where on the right $(-)^\ast$ denotes the continuous linear dual.
First noticed in (Serre).
By the Hirzebruch-Riemann-Roch theorem the index of the Dolbeault operator is the Todd genus.
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
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 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: