This question is inspired by, but is independent of: [Sheaf Description of G-Bundles](https://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles)


Line bundles are classified by $H^1(X,\mathcal{O}^\times\_X)$. We also know that in general that $H^1(X,G)$, where $G$ is a sheaf from open sets in $X$ to $Grps$, classifies $G$-torsors over X.


With this insight in mind: $\mathcal{O}^\times\_X$-torsors should correspond to line bundles. Indeed, if $P$ is one, then $\mathcal{O} \_X \times \_{\mathcal{O} \_X^\times}P$ gives the desired line bundle, and all line bundles are achieved this way (see the question I linked to).


My question is about the more mundane $H^1(X,\mathbb{C})$, which can be thought of as $H^1(X,\underline{\mathbb{C}})$ where $\underline{\mathbb{C}}$ is the constant sheaf $\mathbb{C}$ (which I think of as a sheaf going to $Grps$). These should supposedly correspond to $\mathbb{C}$-bundles over $X$. Which appears to be line bundles. But of course there's no reason for $H^1(X,\mathbb{C})$ to equal $H^1(X,\mathcal{O}^\times\_X)$... My intuition is that this should correspond to the more naive version of fiber bundles that doesn't involve a structure group. Do you have any thoughts?



Confusion about how the first cohomology classifies torsors
The general principle is: if you have some objects which are locally trivial but globally possibly not trivial then the isomorphism classes of such objects are classified by $H^1(X,\underline{Aut})$, where $\underline{Aut}$ is the sheaf of automorphisms of your objects.


So, if your objects locally are $U\times \mathbb A^n$ (i.e. vector bundles) or they are $\mathcal O\_U^{\oplus n}$ (i.e. locally free sheaf) then either of these are classifed by $H^1(X, GL(n,\mathcal O))$. For $n=1$, you get $GL(1,\mathcal O)=\mathcal O^\*$. 


Now what is $\mathbb C$ an automorphism group of? Certainly not of line bundles (zero has to go to zero).


