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