# Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice reference, listing (some of) their invariants? If not, what are the incomplete results?

Edit: I learned that this is a hard open problem. Therefore i would like to restrict the question to the case $n=3$, i.e. "What are the (possibly incomplete) results on classification of vector bundles on $\mathbb{P}_{\mathbb{C}}^3$?"

Thanks!

• Note that the classification of complex line bundles isn't special to $\Bbb C P^n$: complex line bundles are always classified by their first Chern class $c \in H^2(X; \Bbb Z)$ (at least for smooth manifolds, I don't know much algebraic geometry). – Henry T. Horton Apr 15 '13 at 22:10
• @Henry T. Horton: that's no longer true for algebraic isomorphism of line bundles. These are classified by the group Pic(X); it has a map to H^2 (first Chern class) but the kernel, written Pic^0(X), can be huge: in fact, it is itself an algebraic variety, of dimension equal to half of b_1(X). – user64687 Apr 15 '13 at 22:16
• @HenryT.Horton Well, i read this earlier today and i think that must be a rougher classification then the algebraic one. For example if one takes an algebraic surface $S$ with $h^1(S, \mathcal{O}_S) >0$ and $h^2(S, \mathcal{O}_S) =0$ then the natural map $Pic(S) \rightarrow H^2(S;\mathbb{Z})$ of line bundles to their Chern class is surjective but has a nontrivial kernel (which is in fact a $h^1(S, \mathcal{O}_S)$-dimensional torus) – Joachim Apr 15 '13 at 22:17
• Oh hahaha @AsalBeagDubh thanks for your post, as you can tell i was writing my comment at the same time as you. Thanks for clearing that mystery up immediately! – Joachim Apr 15 '13 at 22:18
• @Joachim: great minds... By the way, I don't have time to answer now, but the short version is "definitely not": for example, it's an open conjecture whether every rank-2 bundle on $\mathbf{P}^n$ for $n \geq 6$ is a sum of line bundles. Google "Hartshorne's conjecture" to get started. – user64687 Apr 15 '13 at 22:22