I am trying to understand the definition of direct images of sheafs in Hitchin lecture notes. Here $f:\tilde{M} \rightarrow M$ is a holomorphic map between Riemann surfaces. My problem appears in the proof for (2) in the case of not regular values.
I don't understand why a section of $L$ has then to look this way.
My thoughts: we can say $h(z)=h(f^{-1}(w))=h((z^kg(z))^{-1}(z^k))$. Is that the correct ansatz?