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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!

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  • $\begingroup$ 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). $\endgroup$ – Henry T. Horton Apr 15 '13 at 22:10
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    $\begingroup$ @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). $\endgroup$ – user64687 Apr 15 '13 at 22:16
  • $\begingroup$ @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) $\endgroup$ – Joachim Apr 15 '13 at 22:17
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    $\begingroup$ 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! $\endgroup$ – Joachim Apr 15 '13 at 22:18
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    $\begingroup$ @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. $\endgroup$ – user64687 Apr 15 '13 at 22:22
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This doesn't answer your question, but maybe is worth a look. This paper classifies vector bundles on smooth affine threefolds. The methods are highly sophisticated.

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