# Structure of vector bundles over $(\mathbb P^1)^r$

Fix an algebraically closed field $$k$$ of characteristic zero. I would like to know a classification of all vector bundles on $$(\mathbb P_k^1)^r$$, or at least of semistable vector bundles.

A theorem of Grothendieck states that a vector bundle $$E$$ of rank $$m$$ on $$\mathbb P^1$$ decomposes as a direct sum of line bundles: $$E \cong \mathcal O(a_1) \oplus \dots \oplus \mathcal O(a_r)$$.

Let $$X$$ be the $$r$$-fold fibre product of $$\mathbb P^1$$'s. We know (Hartshorne III Ex 12.6) that $$\operatorname{Pic}(X) = \mathbb Z^r$$, so every line bundle on $$X$$ is of the form $$\pi_1^*\mathcal O_{\mathbb P^1}(a_1) \otimes \cdots \otimes \pi_r^*\mathcal O_{\mathbb P^1}(a_r)$$, with $$\pi_i$$ the projection maps.

For $$r=2$$, we can find vector bundles of rank $$2$$ by computing $$\operatorname{Ext}^1(\pi_i^*\mathcal O_{\mathbb P^1}(a_i), \pi_j^*\mathcal O_{\mathbb P^1}(a_j))$$, which is non-trivial if $$a_i \leq -2$$ and $$a_j \geq 0$$ or if $$a_i \geq 0$$ and $$a_j \leq -2$$.

Does every rank $$2$$ vector bundle on $$(\mathbb P^1)^2$$ arise as such an extension? What about vector bundles of rank $$2$$ on $$(\mathbb P^1)^r$$ for $$r> 2$$? What about bundles of higher rank? What do vector bundles corresponding to non-trivial extensions look like in terms of transition functions?

By the Harder-Narasimhan filtration, every vector bundle on a projective scheme arises as an iterated extension of semistable vector bundles. Of the line bundles which arise as direct sums of tensor products of pullbacks of line bundles on $$\mathbb P^1$$ as above, it is not hard to show that the only semistable such are those of the form $$\bigoplus_{i=1}^m (\pi_1^*\mathcal O_{\mathbb P^1}(a_1) \otimes \cdots \otimes \pi_r^*\mathcal O_{\mathbb P^1}(a_r))$$. Are there any other semistable vector bundles on $$X$$?

Vector bundles on these are much more complicated. Consider $$r=2$$. We have a two to one map to the projective plane and we can pull back vector bundles from the plane. If $$E,F$$ are bundles on the plane and they become isomorphic when you pull back, then $$E\oplus E(-1)$$ and $$F\oplus F(-1)$$ must be isomorphic.