Question

one can show that the relation between first Chern class and second Chern class of $CP^n$ is

$\frac{2(n+1)}{n} c_2 (M)=c_1 (M)^2$

here $c_1 (M)^2=c_1 (M)∧c_1 (M)$. So is there any recurrence formula for i-th Chern class of $CP^n$ ?

Answer 1

There is no need for a recurrence formula. If $h\in \mathrm H^2(\mathbb C\mathbb P^n, \mathbb Z)$ is the Poincaré dual of a hyperplane section, then $h^i$ generates $\mathrm H^{2i}(\mathbb C\mathbb P^n, \mathbb Z)$ for all $i = 0, \dots, n$. Then it follows from the Euler sequence that $$ \mathrm c_i(\mathbb C\mathbb P^n) = \binom{n+1}{i}h^i. $$