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It is obvious that the sheaf of holomorphic functions is not soft, not every holomorphic function on a closed set (appropriately defined) can be extended to be an entire function. What about the sheaf of non-vanishing holomorphic functions $\mathcal{O}_X^{\ast}$? I assume that it is not soft either, but I'd like an example.

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The case of non-vanishing holomorphic functions is really only easier. For instance, consider the function $f(z)=z$, defined on a non-discrete closed set that does not contain $0$. Since a holomorphic function on a connected set is determined by its values on any non-discrete set, the only possible extension to all of $\mathbb{C}$ is $f(z)=z$, but this is not a non-vanishing entire function.

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