Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic structure? In the case of $CP^N$ the answer is yes and no respectively? So does this generalise?



Holomorphic Line Bundles over a Homogeneous Space
The statement for $\mathbb{CP}^n$ generalizes if $G$ is a linear algebraic group. Then assuming $G/H$ is compact, it is even a projective homogeneous variety, given as quotient of reductive group modulo a parabolic subgroup. 


There are two consequences: 


* first, the quotient $G/H$ is in fact projective, and we can use GAGA to find that the line bundle classifications are the same for holomorphic and algebraic. In particular, we can classify holomorphic line bundles by the Chow group $CH^1(G/H)$ of divisors modulo rational equivalence.
* second, $G/H$ has an algebraic cell structure, which implies that the cycle class map $CH^1(G/H)\to H^2(G/H,\mathbb{Z})$ is an isomorphism.


Taking these two together, we get that the continuous, holomorphic and algebraic line bundle classifications agree.


As pointed out by Francesco Polizzi, this does not generalize to complex tori (algebraic or not). It also does not generalize to higher rank vector bundles.


