Dolbeault cohomology

The Dolbeault cohomology is a mathematical construction in the field of differential topology and complex geometry. It was named after the mathematician Pierre Dolbeault, who defined it in 1953 and examined. The Dolbeault cohomology is a special cohomology theory. As an analogue of the De Rham cohomology on complex manifolds it is also centrally located in the Hodge theory.

Dolbeault complex

The following will be denoted by the set of differential forms. Be one -dimensional complex manifold, an open subset and

Dolbeault the cross - operator. Then say the sequence

P-th Dolbeault complex. This complex is a Kokettenkomplex because it applies Since the underlying manifold is finite, the complex breaks off after steps. Furthermore, the complex is Dolbeault elliptical, that is, the symbols of the main Kokettenkomplex is exact.

Dolbeault cohomology

For this p-th Kokettenkomplex obtained in the usual way a cohomology. This cohomology is called p-th Dolbeault cohomology and is listed by. The q -th cohomology group of the p-th Dolbeault cohomology or the short -th Dolbeault group is thus defined as

Just as with the de Rham cohomology, the cohomology groups are vector spaces.

Set of Dolbeault

The set of Dolbeault is a complex analogue of the theorem of de Rham. With the sheaf of holomorphic p- forms is called on the complex manifold. The set of Dolbeault now states that the qth Garbenkohomologiegruppe with values ​​in the holomorphic p- forms is isomorphic to the q-th cohomology group of the p-th Dolbeault cohomology is. In mathematical brevity, this means