I have the following question:


I know that every holomorphic function $f$ defined on a closed complex submanifold $M$ of the space $\mathbb C^d$ can be extended to a holomorphic function on the total space $\mathbb C^d$.


Now, what about functions which are only defined on an open subset $U$ of $M$? Is it possible to extend them to an open subset of $\mathbb C^d$ containing $U$ ?


I know the corresponding statements are true for real smooth functions since we may une smooth partitions of unity but for holomorphic functions this obviously does not work...


I would be very thankful for any ideas on this subject.


Thanks in advance, 
Tom



Continuations of holomorphic functions on submanifolds to the total space
The construction as suggested by Loïc Teyssier can be globalized, with the help of a Stein neighborhood basis.
By Corollary 1 in ''Every Stein subvariety admits a Stein neighborhood'' by Siu, every Stein 
manifold $M$, which is a submanifold of some manifold $N$ has a neighborhood $W$ in $N$ such that there exist a holomorphic retraction $r : W \to M$. In particular, when $N$ is Stein (as is the case here, with $N = \mathbb{C}^d$), then also $M$ is Stein, and the theorem applies.


Thus, if $U \subseteq M$ is open, $f : U \to \mathbb{C}$ is holomorphic, and one lets $V$ be the open set $V := r^{-1}(U) \subseteq W \subseteq \mathbb{C}^d$, then $F = f \circ r : V \to \mathbb{C}$ is an extension of $f$.


