Is it possible for each vector bundle to find another one that their direct sum is trivial on an affine?

Question

Given a smooth projective variety $X$ and a hyperplane section complement $U$. For any vector bundle $E$ on $X$, is it possible to find another vector bundle $F$ such that $E\oplus F$ is a trivial vector bundle on the affine scheme $U$?

Answer 1

The bundle $E(n)$ is globally generated for $n \gg 0$. Therefore, there is an exact sequence $$ 0 \to F \to \mathcal{O}_X^{\oplus m} \to E(n) \to 0. $$ Since $\mathcal{O}_X(1)\vert_U \cong \mathcal{O}_U$, its restriction to $U$ takes the form $$ 0 \to F\vert_U \to \mathcal{O}_U^{\oplus m} \to E\vert_U \to 0. $$ Finally, $$ \mathrm{Ext}^1(E\vert_U,F\vert_U) = H^1(U,E^\vee \otimes F\vert_U) = 0 $$ since $U$ is affine, hence the second exact sequence splits, hence $E\vert_U \oplus F\vert_U$ is trivial.