I have this question coming from an earlier [Qiaochu's post](https://mathoverflow.net/questions/5243/why-is-it-a-good-idea-to-study-a-ring-by-studying-its-modules). Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just wondering, is there any big reason why the study of bundles would give information about varieties? (I suppose that for this matter, I should actually replace varieties by manifolds?)


I have heard about some invariants, like the Picard group for complex manifolds. But given my inexperience in these concepts, I don't really know why they should be important. So for those who are thinking about "cooking up invariants", some more detailed explanations on why they are useful (and hopefully some elementary examples!) would be appreciated. Thanks!



Why is it useful to study vector bundles?
Well, in algebraic geometry, here's a couple of reasons:


1) Subvarieties: Take a vector bundle, look at a section, where is it zero? Lots of subvarieties show up this way (not all, see [this question](https://mathoverflow.net/questions/1614/when-is-a-scheme-a-zero-set-of-a-section-of-a-vector-bundle)) but generally, we can get lots of information out of vector bundles regarding subvarieties.


2) Invariants of spaces: The Picard group of Line bundles and more generally the Grothendieck group/ring is a useful invariant for differentiating spaces and analyzing the geometry indirectly. On smooth spaces, in fact, complexes of vector bundles can be used to replace coherent sheaves entirely (I believe by the Syzygy Theorem).


3) Maps into Projective Space: This one is line bundle specific. Let $V\to\mathbb{P}^n$ be any imbedding, say, then the pullback of $\mathcal{O}(1)$ is a line bundle on $V$. The nice thing is, the global sections of this line bundle determine and are determined by the map (we can get degenerate mappings by taking subspaces, but lets ignore that, and base loci for the moment). It turns out that we can define a line bundle to be ample, a condition just on the bundle, and that suffices to say that a power of it gives a morphism to $\mathbb{P}^n$, so understanding maps into projective space is the same thing as studying ample line bundles on a variety.


Hope that helps, there's a lot more, but those are the first three things that came to mind.


