This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems straightforward to show surjectivity of stalks by invoking the fact that the logarithm exists for simply connected components. I'm wondering if there is a more categorical approach I'm missing?
2026-03-28 06:40:48.1774680048
Exact sequence of sheaves of holomorphic functions
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Don't use stalks, use the following characterization of epis of sheaves: Every section in the target admits a covering such that each restricted section has a preimage in the source. Here, you only need the trivial direction that such a morphism is an epi. Therefore, the existence of local logarithms is precisely what we need. You cannot get this for free by abstract nonsense, because somewhere we really have to use that we are looking at the exponential function.