Complex differential form

A complex differential form is a mathematical object from the complex geometry. A complex differential form is an analogue of (real) differential forms on complex manifolds. Just as in the real case also form the complex differential form a graded algebra. A complex differential form of degree (or briefly k- form) can be decomposed in a clear manner in two differential forms, which then have the degree or with. To emphasize this decomposition, one also speaks of (p, q) - forms. In this short speech is also clear that there are complex differential forms, for real forms have no such decomposition. An important role is played by the complex calculus of differential forms in Hodge theory.

Complex differential forms

Let be a complex manifold of (complex) dimension. Choose

As a local base of the complexified Kotangentialraums. The covectors have the local representation

The rooms are which only use basis vectors of the form verbally (1,0) - forms and a formula denoted by. Analogously, the space of the (0,1) forms, in other words the covectors having only basic vectors of the form. These two spaces are stable, that is, under holomorphic coordinate change these spaces are mapped into itself. For this reason, the rooms and complex vector bundles over.

Using the outer product of complex differential forms, which is defined exactly as for real differential forms, you can now the spaces of forms by

Define. Next we define the space more than the direct sum

The shapes with. This is isomorph to a direct sum of the spaces of real differential spaces. Also, for a projection

Defined which assigns to each complex differential form of degree their decomposition.

A form has thus in local coordinates that uniquely representation

Since this representation is still very long, it is customary shorthand

To agree.

Dolbeault operators

Definition

The exterior derivative

Which is synonymous with

Can be split into. The Dolbeault operators

And

Are defined by

In local coordinates this means

And

In this case, and on the right side of the equation, the normal Dolbeault operators.

Holomorphic differential forms

Meets a differential form of the equation, then one speaks of a holomorphic differential form. In local coordinates can these forms by

Represent, with holomorphic functions. The vector space of holomorphic forms on is quoted at.

Properties

• For these operators applies a Leibniz rule. Let and, then applies

• From the identity

• Be a Kählermannigfaltigkeit, ie a complex manifold with a compatible Riemannian metric, one can form the adjoint Dolbeault cross - operator with respect to this metric. The operator is then a generalized Laplace operator. Applies this operator in the (complex) Hodge theory.