Functions on an algebraic subvariety X of A^n are the same as functions on A^n restricted to X. So the statement that functions on X extend to all of A^n follows by the definition. My question is: does the analogous statement hold for C^n and closed complex submanifolds (maybe even closed analytic subvarieties), and if so, how is this proved?
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Yes, this is true. It follows from "Cartan's Theorem B" which says that H^1 of any coherent analytic sheaf on a closed submanifold of C^n is 0; the same result is also true for analytic subspaces. Look up any book on several complex variables for a proof. (It is quite possible that there is a more elementary proof.)
(One uses the theorem as follow: Let X be the submanifold or analytic space and consider the exact sequence of sheaves on C^n
0 --> I --> O_{C^n} --> O_X --> 0
where I is the ideal sheaf of X. The vanishing of H^1(C^n,I) implies that the map H^0(C^,O_{C^n}) to H^0(X,O_X) is surjective, which is what you want.)
