PlanetMath: $\bar{\partial}$ operator

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$\bar{\partial}$ operator

(Definition)

Let $G \subset {\mathbb{C}}^n$ be a domain and let $f \colon G \to {\mathbb{C}}$ be a function (continuously differentiable) $(z^1,\ldots,z^n) \mapsto f(z^1,\ldots,z^n)$ where . We can think of as a subset of ${\mathbb{R}}^{2n}$ . We therefore have the following partial derivatives for all $1 \leq j \leq n$ ,

$\displaystyle \frac{\partial f}{\partial z^j}$	$\displaystyle := \frac{1}{2} \left( \frac{\partial f}{\partial x^j} - i \frac{\partial f}{\partial y^j} \right) ,$
$\displaystyle \frac{\partial f}{\partial \bar{z}^j}$	$\displaystyle := \frac{1}{2} \left( \frac{\partial f}{\partial x^j} + i \frac{\partial f}{\partial y^j} \right) .$

Now let

be the standard exterior derivative on ${\mathbb{R}}^{2n}$ and the

and

the standard basis of cotangent vectors. Then if we define

$\displaystyle dz^j$	$\displaystyle := dx^j + i dy^j ,$
$\displaystyle d\bar{z}^j$	$\displaystyle := dx^j - i dy^j ,$

then we can define two new operators acting on

functions on

giving 1-forms by

$\displaystyle \partial f$	$\displaystyle := \sum_{j=1}^n \frac{\partial f}{\partial z^j} dz^j ,$
$\displaystyle \bar{\partial} f$	$\displaystyle := \sum_{j=1}^n \frac{\partial f}{\partial \bar{z}^j} d\bar{z}^j .$

By direct calculation we immediately see that

$\displaystyle df = \partial f + \bar{\partial} f .$

Similarly we now define $\partial$ and $\bar{\partial}$ on arbitrary differential form $\omega = \sum_{\alpha,\beta} f_{\alpha,\beta} dz^\alpha \wedge d\bar{z}^\beta$ , where $\alpha$ and $\beta$ range over all multi-indices with elements less then , where if $\alpha = (\alpha_1,\ldots,\alpha_k)$ then $dz^\alpha = dz^{\alpha_1} \wedge \ldots \wedge dz^{\alpha_k}$ , and $f_{\alpha,\beta}$ is a , complex valued function on .

$\displaystyle \partial \omega$	$\displaystyle := \sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial z^j} dz^j \wedge dz^\alpha \wedge d\bar{z}^\beta ,$
$\displaystyle \bar{\partial} \omega$	$\displaystyle := \sum_{\alpha,\beta} \frac{\partial f_{\alpha,\beta}}{\partial \bar{z}^j} d\bar{z}^j \wedge dz^\alpha \wedge d\bar{z}^\beta .$

Again a direct calculation shows that $d = \partial + \bar{\partial}$ .

The Cauchy-Riemann equations are then given by

$\displaystyle \bar{\partial} f = 0$

That is,

is holomorphic if and only if it satisfies the above equations. Note that this only applies to functions. If $\bar{\partial}\omega = 0$ for a differential form, then the coefficients in the standard basis need not be holomorphic.

Proposition 1 $\bar{\partial}$ and $\partial$ satisfy the following properties

$\bar{\partial}$ and $\partial$ are linear,
$\bar{\partial}^2 = \bar{\partial} \bar{\partial} = 0$ and $\partial^2 = \partial \partial = 0$ ,
$\bar{\partial} \partial - \partial \bar{\partial} = 0$ .

While $\bar{\partial} u = 0$ is our condition for to be a holomorphic function it turns out that it is more important to solve the inhomogeneous $\bar{\partial}u = f$ equation, as that allows us to construct holomorphic objects from nonholomorphic ones.

Bibliography

1: Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
2: Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

" $\bar{\partial}$ operator" is owned by jirka.

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Other names:

d bar operator, d-bar operator

Also defines:

$\partial$ operator

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Cross-references: objects, inhomogeneous, properties, coefficients, equations, holomorphic, Cauchy-Riemann equations, complex, multi-indices, range, differential form, 1-forms, operators, vectors, cotangent, standard basis, exterior derivative, partial derivatives, subset, continuously differentiable, function, domain
There are 6 references to this entry.

This is version 4 of $\bar{\partial}$ operator, born on 2005-04-05, modified 2005-11-03.
Object id is 6931, canonical name is BarpartialOperator.
Accessed 3429 times total.

Classification:

AMS MSC:	30E99 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Miscellaneous)
	32A99 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Miscellaneous)

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