Let $L \to X$ be a holomorphic line bundle over a compact complex manifold. Suppose $L$ is non-trivial and has no non-trivial sections. Let me ask the following (hopefully not entirely trivial) question:
Does the dual $L^{\ast}$ have a non-trivial section?
A special case of this is when $L$ is the dual of an ample line bundle. Obviously ample line bundles have sections, but the dual does not.