TITLE: Chern numbers of Projective Space
QUESTION [9 upvotes]: Consider the $k$-th chern class $c_k:=c_k(\mathcal{T}_{\mathbb{P}^n})$ of the tangent sheaf of projective space $\mathbb{P}^n=\mathbb{P}^n_\Bbbk$ over some (algebraically closed, if you want) field $\Bbbk$. I am then wondering what the degree of $\prod_{k=1}^n c_k^{\nu_k}$ is, given that $\sum_{k=1}^n k\nu_k=n$. For instance, I would already be happy to see how to compute $c_2c_1^{n-2}$.
This is certainly well-known, but I cannot find a good reference for it and I am not quite comfortable yet computing it myself, I am looking for a good example to get some more "hands-on" experience with chern numbers.

REPLY [9 votes]: On $\mathbb{P}^n$, there is the so-called Euler sequence - a short exact sequence of vector bundles
$$ 0\to \mathcal{O}_{\mathbb{P}^n}\to \mathcal{O}_{\mathbb{P}^n}(1)^{n+1} \to \mathcal{T}_{\mathbb{P}^n} \to 0. $$
For a short exact sequence of vector bundles $0\to A\to B\to C\to 0$, we have the equality $c(B) = c(A)\cdot c(C)$ where $c$ is the total Chern class:
$$ c(E) = 1 + c_1(E) + c_2(E) + \ldots $$
Let $H\in H^2(\mathbb{P}^n)$ be the class of a hyperplane. Then $c(\mathcal{O}_{\mathbb{P}^n}(1)) = 1+H$, so 
$$ c(\mathcal{T}_{\mathbb{P}^n}) = c(\mathcal{O}_{\mathbb{P}^n})/c(\mathcal{O}_{\mathbb{P}^n}(1)) = 1/(1+H)^{n+1}. $$
This tells you that $c_k(\mathcal{T}_{\mathbb{P}^n}) = (-1)^k \binom{n+1}{k}H^k$. The cohomology groups $H^{2k}(\mathbb{P}^n)$ are free spanned by $H^k$ for $k\leq n$ and zero for $k>n$.