Subbundles of a rank-two trivial bundle on $\mathbb{P}^1$.

Question

Can every line bundle $\mathcal{O}(a)$ on $ \mathbb{P}^1$ be realized as a subbundle of the trivial bundle $\mathcal{O}\oplus \mathcal{O}$ on $\mathbb{P}^1$? I know that this phenomena can happen in general: for example Moebius band gives an example of a nontrivial real subbundle of the rank-two real trivial bundle on $S^1$, however I can't think of any meaningful obstruction here.

Answer 1

Yes. For every $k > 0$, we have $\mathcal{O}(k)\oplus\mathcal{O}(-k) \cong \mathcal{O}\oplus\mathcal{O}$ as smooth bundles over $\mathbb{CP}^1$. To see this, note that

$$\operatorname{rank}_{\mathbb{R}}(\mathcal{O}(k)\oplus\mathcal{O}(-k)) = 4 > 2 = \dim_{\mathbb{R}}\mathbb{CP}^1,$$

so $\mathcal{O}(k)\oplus\mathcal{O}(-k)$ admits a nowhere-zero section, and hence $\mathcal{O}(k)\oplus\mathcal{O}(-k) \cong L\oplus\mathcal{O}$ for some complex line bundle $L$. As

$$c_1(L) = c_1(L\oplus\mathcal{O}) = c_1(\mathcal{O}(k)\oplus\mathcal{O}(-k)) = c_1(\mathcal{O}(k)) + c_1(\mathcal{O}(-k)) = 0,$$

and smooth complex line bundles are classified by their first Chern class, we have $L \cong \mathcal{O}$, and hence $\mathcal{O}(k)\oplus\mathcal{O}(-k) \cong L\oplus\mathcal{O} \cong \mathcal{O}\oplus\mathcal{O}$.

Note however that $\mathcal{O}(k)\oplus\mathcal{O}(-k)$ and $\mathcal{O}\oplus\mathcal{O}$ are not isomorphic as holomorphic vector bundles. To see this, note that

$$\Gamma(\mathbb{CP}^1, \mathcal{O}(k)\oplus\mathcal{O}(-k)) = \Gamma(\mathbb{CP}^1, \mathcal{O}(k))\oplus\Gamma(\mathbb{CP}^1, \mathcal{O}(-k)) = \Gamma(\mathbb{CP}^1, \mathcal{O}(k))\oplus\{0\}.$$

As $\Gamma(\mathbb{CP}^1, \mathcal{O}(k))$ can be identified with the set of degree $k$ homogeneous polynomials in two variables, it has dimension $k + 1$.

On the other hand,

$$\Gamma(\mathbb{CP}^1, \mathcal{O}\oplus\mathcal{O}) = \Gamma(\mathbb{CP}^1, \mathcal{O})\oplus\Gamma(\mathbb{CP}^1, \mathcal{O}) = \mathcal{O}(\mathbb{CP}^1)\oplus\mathcal{O}(\mathbb{CP}^1) = \mathbb{C}\oplus\mathbb{C}$$

which has dimension $2$. So for $k \neq 1$, we see that $\mathcal{O}(k)\oplus\mathcal{O}(-k)$ and $\mathcal{O}\oplus\mathcal{O}$ are not isomorphic as holomorphic vector bundles. I think there should be an easier way of demonstrating this fact which would also include the case $k = 1$, but I can't think of it right now (other than appealing to semistability directly).