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Let X be a compact surface and $\tilde{X}$ its blowing-up, how can I show the formula $c_2(\tilde{X})=c_2(X)+1$?

edited Apr 9 2012 at 7:54

Yemon Choi
8,446●1●19●44

asked Apr 9 2012 at 7:50

Marcello
11●1

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For any complex manifold $X$, $c_n(X)=\chi (X)$ ($\chi$ is the Euler characteristics). So you just need to figure out how $\chi$ changes when you blow up (hitn: you replace a point by an $S^2$). – Dmitri Apr 9 2012 at 8:43

I'd prefer to show it by computing transition functions of $T\tilde{X}$ and to prove something like $T\tilde{X}=TX\otimes E$, where E is the exceptional divisor. – Marcello Apr 10 2012 at 8:40

Harry · Answer 1 · 2012-04-09 10:49:53Z UTC

Let me slightly expand on Dmitri's comment.

Let $X$ be a finite type separated $\mathbf{C}$-scheme. Let $e_c(X)$ be the compactly supported Euler characteristic. (We consider the singular cohomology of $X$ with $\mathbf{Q}$-coefficients.)

Then, if $X$ is irreducible and $n$-dimensional, by Gauss-Bonnet, we have $\deg c_n(X) = e_c(X)$.

Let $p:Y\to X$ be a proper birational surjective morphism. Let $s$ be the number of exceptional components of $p$. It is easy to see that $e_c(Y) = e_c(X) +s$.

This generalizes what you need.

Liviu Nicolaescu · Answer 2 · 2012-04-09 13:36:14Z UTC

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Check this question where a more general situation is discussed.

http://mathoverflow.net/questions/92660,

answered Apr 9 2012 at 13:36

Liviu Nicolaescu
11.3k●1●20●33

Second Chern Class of Blow-up

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