Chern numbers of Projective Space

Question

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.

Answer 1

Edit: as implicitly pointed out in the other answer, I messed up when I claimed that Milnor's class was a hyperplane class. It's the negative of a hyperplane class. I think I fixed that below.

Write $a$ for the negative of the cohomology class of a hyperplane. In Milnor's legendary book "Characteristic Classes", theorem 14.10 states that the $k$th Chern class of the tangent sheaf of complex projective space is $$c_k(\mathcal{T}_{\mathbb{P}^n})=\left( \begin{matrix} n+1 \\ k \end{matrix} \right) a^k.$$ Therefore the degree of the class corresponding to the partition of $n$ with multiplicity sequence $\nu$ is $$(-1)^n \prod \left( \begin{matrix} n+1 \\ k \end{matrix} \right)^{\nu_k}.$$ In particular, the degree of $c_2 c_1^{n-2}$ is $$\mathrm{deg}(c_2 c_1^{n-2})=(-1)^n (n+1)^{n-2} \left( \begin{matrix} n+1 \\ 2 \end{matrix} \right). $$

The proof of Milnor's theorem 14.10 is not hard (it was worth it for me to work it out a couple of times on my own): it uses the standard identification of the tangent bundle with the bundle of homomorphisms from the universal (sometimes, "tautological") bundle to its orthogonal complement (this is in fact precisely the same thing as the Euler sequence, so the two answers are basically the same).

Answer 2

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: $$ c(E) = 1 + c_1(E) + c_2(E) + \ldots $$

In particular, $$ c(\mathcal{O}_{\mathbb{P}^n}(1)^{n+1}) = c(\mathcal{O}_{\mathbb{P}^n})c(\mathcal{T}_{\mathbb{P}^n}).$$

Now, $ c(\mathcal{O}_{\mathbb{P}^n}) =1$. On the other hand, if $H\in H^2(\mathbb{P}^n)$ denotes the class of a hyperplane, then $c(\mathcal{O}_{\mathbb{P}^n}(1)) = 1+H$.

This tells you that $$c(\mathcal{T}_{\mathbb{P}^n}) = (1+H)^{n+1}, $$ and that $c_k(\mathcal{T}_{\mathbb{P}^n}) = \binom{n+1}{k}H^k$.

The cohomology groups $H^{2k}(\mathbb{P}^n)$ are free spanned by $H^k$ for $k\leq n$ and zero for $k>n$.