Complex differential form

From Wikipedia, the free encyclopedia

In mathematics, a complex form is a differential form on a complex manifold. In terms of local holomorphic coordinates, a (p,q)-form is the wedge product of p 1-forms dz_i and q 1-forms that are differentials of antiholomorphic coordinate functions. This extends to multiples of those k-forms by a function, with k = p + q. Any k-form is a sum of such (p,q)-forms, where p runs from 0 to k. Furthermore, the decomposition into (p,q) type is independent of the local holomorphic chart.

Where Hodge theory applies, more is known.

1 Complex differential forms
2 Holomorphic forms
3 See also

[edit] Complex differential forms

It is frequently useful to consider differential forms with complex coefficients. One important case is the study of differential forms over a complex manifold M. Recall that this means that there is a local coordinate system consisting of n complex functions z¹,...,zⁿ and such that the coordinate transitions from one patch to another are holomorphic functions of these variables. Because the holomorphic transition condition is much more constrained than the weaker smoothness condition for coordinate transitions on smooth manifolds, the complex differential forms on a complex manifold also carry a richer structure.

[edit] One-forms

We begin with the case of one-forms. First decompose the complex coordinates into their real and imaginary parts: z^j=x^j+iy^j for each j. Letting

$dz^j=dx^j+idy^j,\quad d\bar{z}^j=dx^j-idy^j,$

one sees that any differential form with complex coefficients can be written uniquely as a sum

$\sum_{j=1}^n f_jdz^j+g_jd\bar{z}^j.$

Let Ω^1,0 be the space of complex differential forms containing only $d z$ 's and Ω^0,1 be the space of forms containing only $d\bar{z}$ 's. One can show, by the Cauchy-Riemann equations, that the spaces Ω^1,0 and Ω^0,1 are stable under holomorphic coordinate changes. In other words, if one makes a different choice w_i of holomorphic coordinate system, then elements of Ω^1,0 transform tensorially, as do elements of Ω^0,1. Thus the spaces Ω^0,1 and Ω^1,0 determine complex vector bundles on the complex manifold.

[edit] Higher degree forms

The wedge product of complex differential forms is defined in the same way as with real forms. Let p and q be a pair of non-negative integers ≤ n. The space Ω^p,q of (p,q)-forms is defined by taking linear combinations of the wedge products of p elements from Ω^1,0 and q elements from Ω^0,1. Symbolically,

$\Omega^{p,q}=\Omega^{1,0}\wedge\dotsb\wedge\Omega^{1,0}\wedge\Omega^{0,1}\wedge\dotsb\wedge\Omega^{0,1}$

where there are p factors of Ω^1,0 and q factors of Ω^0,1. Just as with the two spaces of 1-forms, these are stable under holomorphic changes of coordinates, and so determine vector bundles.

If E^k is the space of all complex differential forms of total degree k, then each element of E^k can be expressed in a unique way as a linear combination of elements from among the spaces Ω^p,q with p+q=k. More succinctly, there is a direct sum decomposition

$E^k=\Omega^{k,0}\oplus\Omega^{k-1,1}\oplus\dotsb\oplus\Omega^{1,k-1}\oplus\Omega^{0,k}=\bigoplus_{p+q=k}\Omega^{p,q}.$

Because this direct sum decomposition is stable under holomorphic coordinate changes, it also determines a vector bundle decomposition.

In particular, for each k and each p and q with p+q=k, there is a canonical projection of vector bundles

$\pi^{p,q}:E^k\rightarrow\Omega^{p,q}.$

[edit] The Dolbeault operators

The usual exterior derivative defines a mapping of sections d:E^k→E^k+1. Restricting this to sections of Ω^p,q, one can show that in fact d:Ω^p,q→Ω^p+1,q + Ω^p,q+1. The exterior derivative does not in itself reflect the more rigid complex structure of the manifold.

Using d and the projections defined in the previous subsection, it is possible to define the Dolbeault operators:

$\partial=\pi^{p+1,q}\circ d:\Omega^{p,q}\rightarrow\Omega^{p+1,q},\quad \bar{\partial}=\pi^{p,q+1}\circ d:\Omega^{p,q}\rightarrow\Omega^{p,q+1}$

To describe these operators in local coordinates, let

$\alpha=\sum_{|I|=p,|J|=q}\ f_{IJ}\,dz^I\wedge d\bar{z}^J\in\Omega^{p,q}$

where I and J are multi-indices. Then

$\partial\alpha=\sum_{|I|,|J|}\sum_\ell \frac{\partial f_{IJ}}{\partial z^\ell}\,dz^\ell\wedge dz^I\wedge d\bar{z}^J$

$\bar{\partial}\alpha=\sum_{|I|,|J|}\sum_\ell \frac{\partial f_{IJ}}{\partial \bar{z}^\ell}d\bar{z}^\ell\wedge dz^I\wedge d\bar{z}^J.$

The following properties are seen to hold:

$d=\partial+\bar{\partial}$

$\partial^2=\bar{\partial}^2=\partial\bar{\partial}+\bar{\partial}\partial=0.$

These operators and their properties form the basis for Dolbeault cohomology and many aspects of Hodge theory.

[edit] Holomorphic forms

For each p, a holomorphic p-form is a holomorphic section of the bundle Ω^p,0. In local coordinates, then, a holomorphic p-form can be written in the form

$\alpha=\sum_{|I|=p}f_I\,dz^I$

where the f_I are holomorphic functions. Equivalently, the (p,0)-form α is holomorphic if and only if

$\bar{\partial}\alpha=0.$

The sheaf of holomorphic p-forms is often written Ω^p, although this can sometimes lead to confusion so many authors tend to adopt an alternative notation.

Categories: Complex manifolds | Differential forms

Complex differential form

From Wikipedia, the free encyclopedia

Contents

[edit] Complex differential forms

[edit] One-forms

[edit] Higher degree forms

[edit] The Dolbeault operators

[edit] Holomorphic forms

[edit] See also

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