under construction
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Dolbeault cohomology of a complex manifold $X$ is the chain cohomology of the Dolbeault complex of $X$ (see there for more).
By the Dolbeault theorem (see there) Dolbeault cohomology is equivalently the abelian sheaf cohomology $H^q(X;\Omega_X^p)$, of the abelian sheaf $\Omega_X^p$ is Dolbeault complex of holomorphic p-forms.
For $X$ a Hermitian manifold write $\mathcal{H}^{p,q}(X)$ for the space of $(p,q)$-harmonic differential forms and write $H^{p,q}$ for its Dolbeault cohomology in the bidegree.
There is a canonical homomorphism
If $X$ is compact, then this is an isomorphism, the Hodge isomorphism
Also
is an isomorphism
(e.g. Maddock, prop. 4.2.7)
Serre duality: On a Hermitian manifold $X$ of complex dimension $dim_{\mathbb{C}}(X) = n$ the Hodge star operator induces isomorphisms
(e.g. Maddock, prop. 4.2.8)
If $X$ is a compact Kähler manifold then
(e.g. Maddock, prop. 4.2.9)
This is called the Hodge decomposition.
Over complex manifolds $X$, Hodge symmetry is the property that the Dolbeault cohomology groups $H^{p,q}(X)$ are taken into each other under complex conjugation followed by switching the bidegree:
In particular this means that the dimension of the cohomology groups in degree $(p,q)$ coincides with that in bidegree $(q,p)$.
Given any holomorphic vector bundle $E$, one can form the Dolbeault resolution $E \otimes \Omega^{0,q}$, where $\Omega^{0,q}$ is the sheaf of $C^\infty$ $(0,q)$-forms. This is an acyclic resolution of $E$ and hence computes its sheaf cohomology.
(…)
The Dolbeault complexes naturall fit into a double complex
The corresponding spectral sequence of a double complex is called the Frölicher spectral sequence. On a Kähler manifold it exhibits the Hodge filtration.