Now let be the standard exterior derivative on and the and the standard basis of cotangent vectors. Then if we define
then we can define two new operators acting on functions on giving 1-forms by
By direct calculation we immediately see that
Again a direct calculation shows that .
The Cauchy-Riemann equations are then given by
That is, is holomorphic if and only if it satisfies the above equations. Note that this only applies to functions. If for a differential form, then the coefficients in the standard basis need not be holomorphic.
and satisfy the following properties
and are linear,
While is our condition for to be a holomorphic function it turns out that it is more important to solve the inhomogeneous equation, as that allows us to construct holomorphic objects from nonholomorphic ones.
- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|Date of creation||2013-03-22 15:10:39|
|Last modified on||2013-03-22 15:10:39|
|Last modified by||jirka (4157)|
|Synonym||d bar operator|