I am reading a paper in differential geometry, Hitchin's Langlands duality and G2 spectral curves (see the end of page 8 in the arxiv version), where $f: E \rightarrow F$ is a morphism of holomorphic vector bundles on a Riemann surface, and the kernel bundle $K$ is considered. A standard argument shows that $K$ is a vector bundle too. In the paper, the nature of the map ensures also that $K$ is not a zero bundle.

Anyway, the author considers a proper divisor $D$ on the Riemann surface "where $K$ is null". Does it make any sense?

(I study algebraic geometry so maybe there is some kind of "differential" approach I am missing here, I don't know...)