Are splitting vector bundles closed under kernel or cokernels?

Given an ample line bundle $$L$$ on a smooth projective variety of dimension $$\geq 2$$, let $$C$$ be the category of vector bundles that are direct sums of powers of $$L$$. Two related questions:

1. Given a surjection in $$C$$ does the kernel have a filtration by line bundles? (a filtration that the successive quotients are line bundles)
2. Given an injection in $$C$$ where the cokernel is a vector bundle, does the cokernel have a filtration by line bundles?

If both of these are not true is there any family of varieties of dimension $$\geq 2$$ that any of them are true?

Definitely not. For instance, consider the projective space $$\mathbb{P}^n$$ and the Euler sequence $$0 \to \Omega \to \mathcal{O}(-1)^{\oplus (n + 1)} \to \mathcal{O} \to 0.$$ Its second and third terms are in $$C$$ (if $$L = \mathcal{O}(1)$$), but the first term has no filtration by line bundles. Indeed, the first cohomology of any line bundle on $$\mathbb{P}^n$$ vanishes (when $$n \ge 2$$), hence any such filtration would split, but $$\Omega$$ is indecomposable.