Complex Differential Form

Where Hodge Theory applies, more is known.

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 ''z''¹,...,z^''n'' and such that the coordinate transitions from one patch to another are Holomorphic Function s of these variables. Because the holomorphic transition condition is much more constrained than the weaker smoothness condition for coordinate transitions on Smooth Manifold s, the complex differential forms on a complex manifold also carry a richer structure.

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

dz

's and Ω^0,1 be the space of forms containing only

d\bar{z}

's. By the Cauchy-Riemann Equations , one can show that if we used a different holomorphic coordinate system ''w''^''j'', the spaces Ω^1,0 and Ω^0,1 are stable. Thus they determine complex Vector Bundle s on the complex manifold.

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\dots\wedge\Omega^{1,0}\wedge\Omega^{0,1}\wedge\dots\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\dots\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
ightarrow\Omega^{p,q}.

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

To describe these operators in local coordinates, let

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

	CATEGORIES ABOUT COMPLEX DIFFERENTIAL FORM

	complex manifolds

	differential forms

Information About

Complex Differential Form