# Vector bundle with trivial chern classes

I saw in a paper that the only vector bundle of rank $$r$$ on the projective plane with trivial chern classes is the trivial vector bundle of rank $$r$$.

I can see that the trivial vector bundle has trivial chern classes. How do we prove the converse?

This is not true (probably you missed some assumptions). For instance, take any stable vector budnle $$F$$ with $$c_1(F) = 0$$ and $$c_2(F) = k^2$$ for some $$k \in \mathbb{Z}$$. Take also $$G = \mathcal{O}(kh) \oplus \mathcal{O}(-kh)$$. Now set $$E = F \oplus G$$. Then $$c(E) = c(F)c(G) = (1 + k^2h^2)(1+kh)(1-kh) = 1,$$ so all Chern classes of $$E$$ are trivial. However, $$E$$ is not trivial.