Birkhoff–Grothendieck theorem

Remarks

The idea of the proof: induct on $r$. Choose the largest $a$ with $H^0(\mathcal{E}(-a)) \ne 0$; a section of $\mathcal{E}(-a)$ gives a subbundle $\mathcal{O}(a) \subseteq \mathcal{E}$ whose quotient is a vector bundle of rank $r - 1$, which splits by induction as $\bigoplus \mathcal{O}(b_i)$. The maximality of $a$ forces $b_i \le a$, so the extension class lies in $\bigoplus H^1(\mathbb{P}^1, \mathcal{O}(a - b_i)) = 0$ and the sequence splits. Uniqueness follows from the dimensions $h^0(\mathcal{E}(n))$ for all $n$.

Example: the tangent bundle of $\mathbb{P}^1_k$ is $\mathcal{O}(2)$. The bundles $\mathcal{O}(1) \oplus \mathcal{O}(1)$ and $\mathcal{O} \oplus \mathcal{O}(2)$ both have rank $2$ and degree $2$, but they are not isomorphic. Indeed, after twisting by $\mathcal{O}(-2)$, the first has no nonzero global sections and the second has a one-dimensional space of them.

The statement fails on other curves and in higher dimension. On an elliptic curve there are indecomposable vector bundles of rank $2$, and the tangent bundle of $\mathbb{P}^2$ is not a direct sum of line bundles.

Going deeper

For a student

The integers $a_i$ are called the splitting type of $\mathcal{E}$. The theorem is used to study vector bundles on $\mathbb{P}^n$ by restricting them to lines, where they split, and seeing how the splitting type varies. In the analytic setting it says that every holomorphic vector bundle on the Riemann sphere is a sum of line bundles. This is equivalent to a factorization theorem for matrices of functions on a circle.