I have some problem with integration of differential forms on algebraic surfaces (I'm reading Cartan's book on analytic functions). Let $X \subseteq \mathbb{C}^2$ be an algebraic curve given by polynomial equation $z_1^3 + z_2^3 = 1$. Differentiating this equation we obtain $z_1^2 dz_1 + z_2^2 dz_2 = 0$. We define a form $\omega$ in neighborhood of any point with $z_2 \neq 0$ as $ \omega = \frac{dz_1}{z_2}. $ In neighborhood of points with $z_1 \neq 0$ we define $ \omega = -\frac{z_2dz_2}{z_1^2}. $ This form is defined correctly in the sense that definitions agree if both $z_1 \neq 0$ and $z_2 \neq 0$. I don't know how to integrate such form over some path on my surface, but I know how integrate forms over paths in $\mathbb{C}$. Please help me to understand integration over paths on surfaces within this example or give me some reference. (I did't find any example myself, only nude theory).
Problem with integration of $1$-form on surface
2
$\begingroup$
complex-analysis
riemann-surfaces
differential-forms
1 Answers
1
In general you break up the path into pieces which are contained in a chart, so you can integrate each piece in a specific chart (the same way you integrate forms over paths in $\mathbb{C}$), and then you add up the results from all the pieces.
-
0Yes, that would probably be the easiest way in that case. – 2012-10-30