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:
- Given a surjection in $C$ does the kernel have a filtration by line bundles? (a filtration that the successive quotients are line bundles)
- 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?