Relation among Chern classes

Question

Let $E\rightarrow\mathbb{P}^2$ be a rank $r$ vector bundle over $\mathbb{P}^2$ with Chern classes $c_i = c_i(E)$.

Is there any relation among $c_1$ and $c_2$ or are they in general completely unrelated?

Thank you.

Answer 1

One may consider $(c_1,c_2)$ as a pair of integers by choosing generators of $H^{2}$ and $H^4$ (both isomorphic to $\mathbb{Z}$).

Then it was proved by Schwarzenberger that any pair $(c_1,c_2) \in \mathbb{Z}^2$ is realsed by some rank 2 vector bundle on $\mathbb{CP}^2$. See http://www.numdam.org/article/SB_1978-1979__21__80_0.pdf page 81.

Then taking products with trivial bundles shows that any pair can appear for higher ranks also.

For rank 1 i.e. line bundles then $c_2=0$, but any $c_1$ can occur (as $\mathcal{O}(k)$ shows).

For $\mathbb{CP}^k$ for $k>2$ there are some relations in the Chern classes of suitably high rank vector bundles, see the above reference paper (also page 81).