MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 4 down vote accepted

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.)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.