Reference request: second Chern class of P^2

Question

I have heard that $c_2(T_{\mathbb{P}^2})=e(\mathbb{P}^2)$. What's the general result and where can I read about it? Thanks.

Answer 1

The general result is that for a complex compact manifold $M$ of dimension $n$ we have $$\chi_\text{top}(M)=c_n(T_M) $$ Here $\chi_\text{top}(M)$ denotes the topological Euler class of $M$, the alternating sum $\sum_{i=0}^{2n}(-1)^ib_i$ of its Betti numbers $b_i=\operatorname {dim}_\mathbb Q H^i(M,\mathbb Q)$.
And $c_n(T_M)\in H^{2n}(M,\mathbb Z)=\mathbb Z$ is the top Chern class of the holomorphic tangent bundle of $M$.

It is remarkable that the left-hand side depends only on the topological space underlying $M$ while the right-hand side takes into account the holomorphic structure of $M$.
Griffiths-Harris prove this theorem on page 416 of their book, and call it Gauss-Bonnet Formula III. The theorem also goes under the name Poincaré-Hopf.
In your case ($M=\mathbb P^2_\mathbb C$) you will obtain the formula $1-0+1-0+1=3$, which can also be checked in a more elementary way (ask your little niece) .