Talk:Hirzebruch–Riemann–Roch theorem

Last equation

I like this article, although I don't know much about algebraic geometry. I found this article to be understandable until the last equation. This is in the section about surfaces, where R.e.b. relates the holomorphic Euler characteristic of D to that of the zero divisor. I believe the answer, but some magic seems to have occured to get it from the previous step. Somehow a first Chern class of the tangent bundle has been replaced by the canonical class. I know that for smooth Kahler manifolds these both vanish at the same time, are they really just equal? Or is it something more subtle that happened here?

Somewhat less magically the second Chern character ch_2((O(D)) turned into a product D^2. I guess I should be able to understand this from the comment above about multiplication, but it seems like there's a step missing here too, but something trivial like that c_1(O(D)) is Poincare dual to D, which really seems pretty clear, so you get intersection numbers ... I guess they are equal to the intersection numbers in the Picard group? Thanks.JarahE 00:31, 26 April 2006 (UTC)