Let $X$ be a non-singular complex affine variety in $\mathbb{C}^n$ and let $$f : X \to \mathbb{C}$$ be a holomorphic function. Does $f$ extend to a holomorphic function $f : \mathbb{C}^n \to \mathbb{C}$?
This question is inspired by this one, where the answer is shown to be yes in the special case where $$ X = \{(z, w) \in \mathbb{C}^2 : z^2 + w^2 = 1\}.$$
The answer is yes.
First, the ideal sheaf $I_X$ of $X$ is a coherent analytic sheaf. So there is an exact sequence of coherent analytic sheaves
$$0\to I_X\to \mathcal{O}_{\mathbb C^n}\to \mathcal{O}_X\to 0.$$
Cartan's theorem B implies that $H^1(I_X)=0$. So there is surjectivity
$$H^0(\mathcal{O}_{\mathbb C^n})\twoheadrightarrow H^0(\mathcal{O}_X).$$