I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle $F$ on the projective line $\mathbb{P}^1$ is a direct sum of $r$ line bundles, where $r$ is the rank of $F$.


My question is: to what extent is this true for $\mathbb{P}^1\times\mathbb{P}^1$, if at all? At first glance I imagine this is classical, but I haven't had luck working it out, nor do I have a good reference for where this might be done.


Irrespective of the content of the answer, would it follow that the answer would be the same if the question were asked for $(\mathbb{P}^1)^k$?


Thanks!



Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$
The splitting theorem is most certainly false for vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$. In fact, the theory of vector bundles on quadric surfaces is probably as complicated as the theory of vector bundles on $\mathbb{P}^2$ (that is, very complicated).


Here is a simple example of an indecomposable rank 2 bundle. By the Künneth formula, we see that $\mathrm H^1(\mathcal O(1,-2))$ is not 0; hence there is a non-split extension
$$
0 \to \mathcal O(1,-2) \to E \to \mathcal O \to 0.
$$
I claim that $E$ is not split. Suppose that it splits as the sum $L\_1 \oplus L\_2$ of two line bundles. If $L\_1$ were to map to 0 in $\mathcal O$, the map $E \to \mathcal O$ would factor through $L\_2$, and then the map $L\_2 \to \mathcal O$, being surjective, would have to be an isomorphism, and the sequence would split. Hence both $L\_1$ and $L\_2$ admit a non-zero map to $\mathcal O$; this means that they have to be of the form $\mathcal O(m,n)$ with $m,n \leq 0$. But $L\_1\otimes L\_2$, which is the determinant of $E$, is $\mathcal O(1,-2)$, and this gives a contradiction.


I know that there have been people studying the moduli theory of vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$(see for example [arXiv:0810.4392](https://arxiv.org/abs/0810.4392 "Sukmoon Huh: Moduli of Stable Sheaves on a Smooth Quadric in P_3"); but there must be lots of earlier papers).


