Let $M$ be a complex manifold and $U\subset M$ a domain.

Question: Is every derivation of the complex algebra of holomorphic functions $\mathcal{O}(U)$ induced by a holomorphic vector field defined on $U$?

I think this is true if $M$ is Stein, by a similar proof to the smooth case (using Cartan B instead of bump functions), but I cannot see if it holds in the general case.


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