Complex differential form

A complex differential form is a mathematical object from the complex geometry . A complex differential form is a correspondence of the (real) differential forms on complex manifolds . Just as in the real case, the complex differential forms also form a graduated algebra . A complex differential form of degree (or k-form for short) can be broken down into two differential forms in a unique way, which then have the degree or with . To emphasize this decomposition, one also speaks of (p, q) -forms . This short way of speaking also makes it clear that we are dealing with complex differential forms, because real forms have no such decomposition. The calculus of complex differential forms plays an important role in Hodge's theory . ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p+q=k}$

Complex differential forms

Be a complex manifold of the (complex) dimension . Choose ${\displaystyle M}$ ${\displaystyle n}$

${\displaystyle \{\mathrm {d} z^{j}=dx^{j}+idy^{j},\ \mathrm {d} {\bar {z}}^{j}=dx^{j}-idy^{j};\ 1\leq j\leq n\}}$

as a local basis of the complexified cotangent space . The co- vectors have the local representation

${\displaystyle \sum _{j=1}^{n}f_{j}\mathrm {d} z^{j}+g_{j}\mathrm {d} {\overline {z}}^{j}.}$

The spaces in which only basis vectors of the form occur are verbally referred to as (1,0) -forms and formally with . Analogous to this is the space of the (0,1) -forms, i.e. the covectors, which only have basis vectors of the form . These two spaces are stable, which means that these spaces are mapped into themselves under holomorphic coordinate changes. Because of this, the spaces and complex vector bundles are over . ${\displaystyle \textstyle \mathrm {d} z^{j}}$ ${\displaystyle {\mathcal {A}}^{1,0}(M)}$ ${\displaystyle {\mathcal {A}}^{0,1}(M)}$ ${\displaystyle \textstyle \mathrm {d} {\overline {z}}^{j}}$ ${\displaystyle {\mathcal {A}}^{1,0}(M)}$ ${\displaystyle {\mathcal {A}}^{0,1}(M)}$ ${\displaystyle M}$

With the help of the outer product of complex differential forms, which is defined in the same way as for real differential forms, one can now go through the spaces of the forms ${\displaystyle (p,q)}$

${\displaystyle {\mathcal {A}}^{p,q}(M):=\bigwedge _{j=1}^{p}{\mathcal {A}}^{1,0}(M)\wedge \bigwedge _{j=1}^{q}{\mathcal {A}}^{0,1}(M)}$

define. The space is also defined as the direct sum ${\displaystyle {\mathcal {E}}^{r}(M)}$

${\displaystyle {\mathcal {E}}^{r}(M):=\bigoplus _{p+q=r}{\mathcal {A}}^{p,q}(M)}$

of forms with . This is isomorphic to the direct sum of the spaces of the real differential forms. Also is for a projection ${\displaystyle (p,q)}$ ${\displaystyle r=p+q}$ ${\displaystyle {\mathcal {E}}^{r}(M)\cong {\mathcal {A}}^{r}(M)\oplus i{\mathcal {A}}^{r}(M)}$ ${\displaystyle p+q=r}$

${\displaystyle \pi _{p,q}\colon {\mathcal {E}}^{r}(M)\to {\mathcal {A}}^{p,q}(M)}$

defines which assigns its -decomposition to each complex differential form in terms of degree . ${\displaystyle r}$ ${\displaystyle (p,q)}$

A form has the unique representation in local coordinates ${\displaystyle (p,q)}$ ${\displaystyle z_{1},\ldots ,z_{n}}$

${\displaystyle \omega =\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}.}$

Since this representation is very long, it is common to use the short form

${\displaystyle \sum _{I,J}f_{I,J}\mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}:=\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}^{}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}}$

to agree.

Dolbeault operators

definition

The outer derivative

${\displaystyle \mathrm {d} :{\mathcal {E}}^{r}(M)\to {\mathcal {E}}^{r+1}(M),}$

which is synonymous with

${\displaystyle \mathrm {d} :\bigoplus _{p+q=r}{\mathcal {A}}^{p,q}(M)\to \bigoplus _{p+q=r}{\mathcal {A}}^{p+1,q}(M)\oplus \bigoplus _{p+q=r}{\mathcal {A}}^{p,q+1}(M),}$

can be split into. The Dolbeault Operators ${\displaystyle \mathrm {d} =\partial +{\overline {\partial }}}$

${\displaystyle \partial :{\mathcal {A}}^{p,q}(M)\to {\mathcal {A}}^{p+1,q}(M)}$

and

${\displaystyle {\overline {\partial }}:{\mathcal {A}}^{p,q}(M)\to {\mathcal {A}}^{p,q+1}(M)}$

are defined by

${\displaystyle {\begin{aligned}\partial &:=\pi _{p+1,q}\circ \mathrm {d} \\{\overline {\partial }}&:=\pi _{p,q+1}\circ \mathrm {d} .\end{aligned}}}$

This means in local coordinates

${\displaystyle \partial \left(\sum _{I,J}f_{I,J}\mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}\right)=\sum _{I,J}\sum _{l=1}^{n}{\frac {\partial f_{I,J}}{\partial z^{l}}}\mathrm {d} z^{l}\wedge \mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}}$

and

${\displaystyle {\overline {\partial }}\left(\sum _{I,J}f_{I,J}\mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}\right)=\sum _{I,J}\sum _{l=1}^{n}{\frac {\partial f_{I,J}}{\partial {\overline {z}}^{l}}}\mathrm {d} {\overline {z}}^{l}\wedge \mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}.}$

Where and on the right-hand side of the equation are the normal Dolbeault operators . ${\displaystyle \partial }$ ${\displaystyle {\overline {\partial }}}$

Holomorphic differential forms

If a differential form fulfills the equation , one speaks of a holomorphic differential form. In local coordinates you can get these shapes through ${\displaystyle \omega \in {\mathcal {A}}^{p,0}(M)}$ ${\displaystyle {\overline {\partial }}\omega =0}$

${\displaystyle \sum _{I}f_{I}\mathrm {d} z^{I}}$

represent, where are holomorphic functions. The vector space of the holomorphic forms is also noted. ${\displaystyle f_{I}}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle \Omega ^{p}(M)}$

properties

• A Leibniz rule applies to these operators . Be and , then applies ${\displaystyle \omega \in {\mathcal {A}}^{p,q}}$ ${\displaystyle \nu \in {\mathcal {A}}^{r,s}}$

${\displaystyle \partial (\omega \wedge \nu )=\partial \omega \wedge \nu +(-1)^{p+q}\,\omega \wedge \partial \nu }$

and

${\displaystyle {\overline {\partial }}(\omega \wedge \nu )={\overline {\partial }}\omega \wedge \nu +(-1)^{p+q}\,\omega \wedge {\overline {\partial }}\nu .}$

• From identity

${\displaystyle 0=\mathrm {d} ^{2}=(\partial +{\overline {\partial }})^{2}=\partial ^{2}+({\overline {\partial }}\partial +\partial {\overline {\partial }})+{\overline {\partial }}^{2}}$

follows , and , because all three terms are of different degrees. The operators and are therefore suitable for a cohomology theory. This is called Dolbeault cohomology . ${\displaystyle \partial ^{2}=0}$ ${\displaystyle \partial {\overline {\partial }}+{\overline {\partial }}\partial =0}$ ${\displaystyle {\overline {\partial }}^{2}=0}$ ${\displaystyle \partial }$ ${\displaystyle {\overline {\partial }}}$

• Let a Kahler manifold , i.e. a complex manifold with a compatible Riemannian metric , then one can form the adjoint Dolbeault transverse operator with respect to this metric. The operator is then a generalized Laplace operator . This operator is used in the (complex) Hodge theory . ${\displaystyle (M,g)}$ ${\displaystyle g}$ ${\displaystyle {\overline {\partial }}^{*}}$ ${\displaystyle \Delta :={\overline {\partial }}\,{\overline {\partial }}^{*}+{\overline {\partial }}^{*}{\overline {\partial }}}$

literature

• Eric W. Weisstein : Complex Form . In: MathWorld (English).
• Raymond O. Wells: Differential analysis on complex manifolds. Prentice-Hall, Englewood Cliffs NJ 1973, ISBN 0-13-211508-5 .
• Lars Hörmander : An Introduction to Complex Analysis in Several Variables (= North-Holland Mathematical Library 7). 2nd revised edition. North-Holland et al., Amsterdam et al. 1973, ISBN 0-7204-2450-X .