TITLE: Rank 2 vector bundles over $\mathbb CP^2$
QUESTION [7 upvotes]: Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism?

REPLY [3 votes]: As additional comment, We have Cartan-Serre theorem for construction of rank 2 vector bundles on projetive varieties.
Let $X$ be a complex manifold, $L_1, L_2 ∈ Pic(X)$ are line bundles,
$Z ⊂ X$ with $codim_X(Z) = 2$. Under some cohomological conditions, the
sheaf $E$ sitting in 
$$0 \to L_1 \to E \to L_2 \otimes J_Z → o$$
is locally free. Moreover, If $X$ is projective, any rank-2 holomorphic vector bundle can be constructed this way