Holomorphic Line Bundles over a Homogeneous Space

Question

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?

Answer 1

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.

Answer 2

Complex line bundles over $M$ are classified by their first Chern class, which is an element in $H^2(M, \, \mathbb{Z})$.

Since $H^2(\mathbb{P}^n, \, \mathbb{Z})= \mathbb{Z}$, there is a complex vector bundle for every integer $a \in \mathbb{Z}$. From the classification of holomorphic vector bundles on projective spaces, it is well-known that one can equip such a bundle with a unique holomorphic structure, obtaining the holomorphic vector bundle usually denoted by $\mathcal{O}_{\mathbb{P}^n}(a)$. So in this case the answers to your questions are actually yes and no.

This does not generalise. For instance, take as $M$ an elliptic curve (i.e, a complex torus of dimension $1$). Topologically, $M=S^1 \times S^1$, so again $H^2(M, \, \mathbb{Z})= \mathbb{Z}$ and there is a unique complex line bundle with trivial first Chern class, namely the trivial bundle. However, one can equip this bundle with uncountably many different holomorphic structures: take all the bundles of the form $L_p=\mathcal{O}_M(o-p)$, where $o$ is the origin in the group law of $M$ and $p \in M$ is a varying point.