some examples of the Soft sheaves but not fine

Question

As we know,a fine sheaf is also soft.So,I need some examples of the sheaves that are soft but not fine.Can the holomorphic sheaf $\mathcal O(X)$ be one?Any help and comments are accepted.Thanks a lot!

Answer 1

No, the sheaf of holomorphic maps can be not soft!

Example. Let $\displaystyle X=\left\{z\in\mathbb{C}:|z|\leq\frac{1}{2}\right\}$ be the "half unitary disk"; the holomorphic maps \begin{equation*} f(z)=\sum_{n=0}^{+\infty}z^n,\,g(z)=\sum_{n=1}^{+\infty}z^{n!} \end{equation*} can not be extendend to the whole of $\mathbb{C}$, so $\mathcal{O}_{\mathbb{C}}$ is neither soft nor fine. $\triangle$

Considering the constant sheaf $\mathcal{F}$ to $\mathbb{Z}$ on $\mathbb{A}^1_{\mathbb{C}}$ with Zariski topology; $\mathcal{F}$ is flabby but not fine (of course), so it is soft not fine.