Special and general types
Dolbeault cohomology of a complex manifold is the chain cohomology of the Dolbeault complex of (see there for more).
By the Dolbeault theorem (see there) Dolbeault cohomology is equivalently the abelian sheaf cohomology , of the abelian sheaf is Dolbeault complex of holomorphic p-forms.
For a Hermitian manifold write for the space of -harmonic differential forms and write for its Dolbeault cohomology in the bidegree.
is an isomorphism
(e.g. Maddock, prop. 4.2.7)
Serre duality: On a Hermitian manifold of complex dimension the Hodge star operator induces isomorphisms
(e.g. Maddock, prop. 4.2.8)
If is a compact Kähler manifold then
(e.g. Maddock, prop. 4.2.9)
This is called the Hodge decomposition.
Over complex manifolds , Hodge symmetry is the property that the Dolbeault cohomology groups 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 coincides with that in bidegree .
Given any holomorphic vector bundle , one can form the Dolbeault resolution , where is the sheaf of -forms. This is an acyclic resolution of and hence computes its sheaf cohomology.
Double complex and Frölicher spectral sequence
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.
- Zachary Maddock, Dolbeault cohomology (pdf)