Talk:Differential of the first kind
|WikiProject Mathematics||(Rated Stub-class, Low-importance)|
hi , i find it totally inappropriate the math symbols you use please use latex thank you
If you search google books for differentials and "third kind" or "second kind" you find books by Chvalley and Griffiths: they say a meromorphic one-form on a Riemann surface is of the
1) first kind, if it is holomorphic,
2) second kind, if it has zero residues at each pole
3) third kind if it has only simple poles
I'm not an expert in the field but I have few comments to make: 1. Why the article, needs to be titled "Differentials of the first kind", while there is no independent article for the second and the third kind. I think we should change the title to "Abelian Differentials" and put two (or three) sections for each kind. The fact that "Abelian Differential" is redirected to this page is even more reason to do so.
2. The elliptic integral is a special case of hyper elliptic integral and shouldn't be excluded, i.e., degree > 2 and k >= 0.
3. I don't see that integral as the high light of using differentials, while there are essential to cohomology theory for example. I think we have more important stuff to put here.
4. Finally, I agree with above comment, we need to make the definition of the second and third kind obvious in the article instead of a giving a general unrelated description.
I have to give a talk on the differentials of the second kind, after which I'll come back and update the article.Last modified on 10 April 2010, at 03:11