As is well known, the line bundles over \**CP*\*$^1$ are indexed by the integers. My question is how are the line bundles over \**CP*\*$^n$, $n > 1$, and \**Gr*\*$(n,k)$ indexed? Moreover, do there exist any other interesting classifications of line bundles over spaces (I remember something about Atiyah and elliptic curves)?



Indexing the line bundles over a Grassmannian.
Algebraic line bundles on a smooth variety $X$ are classified by the Picard group $Pic(X) = H^1(X, \mathbf O\_X^\*)$. This is an exercise in Hartshorne's book, basically every line bundle is mapped to it's gluing cocycle.
The group $Pic(X)$ is also equal to the group $CH^1(X)$ of divisors modulo rational equivalence. The map sends a line bundle to it's first Chern class.


Now for a projective space Picard group is $\mathbf Z$ generated by $\mathbf O(1)$.
Picard group of a Grassmannian is also $\mathbf Z$. I believe that the generator is a pullback of $\mathbf O(1)$ for a Plucker embedding $Gr(n,k) \subset \mathbf P^N$.


One can prove it as follows: both $\mathbf P^n$ and $Gr(n,k)$ are algebraically cellular, meaning that they consist of pieces isomorphic to affine spaces: for $\mathbf P^n$ this is obvious, and for $Gr(n,k)$ the cells are Schubert cells.


For such varieties Chow groups coincide with cohomology groups and are generated by cells (it's like computing cohomology of CW complex without cells of odd dimension).
Since there is only one cell of complex codimension 1 for projective spaces and Grassmannians, we get $Pic(X) = H^2(X, \mathbf Z) = \mathbf Z$.


EDIT: 


* Actually, a projective space IS a Grassmannian. :)
* As other people commented, the generator of the $Pic(Gr(n,k))$ is the n'th wedge power of the canonical vector bundle of rank n.


