Over rational curve we know that any vector bundle is decomposable to direct sum of line bundles.
In higher dimensions there are examples of indecomposable bundles.
some indecomposable vector bundles have might have proper sub-bundles (all bundles and sub bundles here are in holomorphic category and not topological)
First Question: Over curves, can we have a indecomposable bundle having a proper sub bundle?
For rational curves the answer is negative obviously, what about elliptic curves, and higher genus curves?
Second: same question for Calabi-Yau 3-folds and K3-surfaces? for example for Quintic.
Please provide examples (or give reference) if you know any.
Best Answer
Over a curve any rank $2$ bundle has a rank $1$ subbundle: Choose a subbundle defined over a Zariski dense open set, and then extend it over the missing points by observing that locally the problem of making such an extension is the problem of extending a map into $\mathbb P^1$.