Are splitting vector bundles closed under kernel or cokernels?

Question

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?

Answer 1

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.

Dualizing the sequence you can obtain an example with injection.