# Does the “Ohsawa-Takegoshi theorem without bounds” have a name?

There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:

Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex domain, $\phi$ a plurisubharmonic function on $\Omega$, and $P = \{z \in \Omega:z_n = 0\}$. If $f$ is a holomorphic function on $\Omega \cap P$ (thought of as a domain in $\mathbb{C}^{n-1}$), and $\int_{\Omega \cap P} |f|^2e^{-\phi}$ is finite, then there is an $F$ holomorphic on $\Omega$ with $F\big|_{\Omega \cap P} = f$ and a constant $C$ depending only on the domain such that $$\int_\Omega |F|^2e^{-\phi} \leq C\int_{\Omega \cap P} |f|^2e^{-\phi}$$

There is a much easier theorem that it is always possible to extend a holomorphic function off a complex hyperplane to the entire domain without requiring any sort of $L^2$ bounds.

Does this more basic "hyperplane extension theorem" have a name? It is pretty important (for instance, it is a crucial ingredient in the solution of the Levi Problem in Krantz's book). It is also used in the proof of Ohsawa-Takegoshi, but it is generally just quoted as a fact in the references I have seen.

## 1 Answer

This is often called Cartan-Serre theorem. Actually Serre's part is that the restriction map $\mathcal O(\Omega)\to\mathcal O(\Omega\cap P)$ is surjective if all cohomology groups $H^{0,p}(\Omega)$ vanish. Now Cartan "B" theorem says that this is the case if the domain $\Omega$ is pseudoconvex. Hence the combination of authors gives the name of the statement. Actually the Cartan-Serre theorem is far more general statement in particular it follows that any holomorphic function on a closed complex submanifold, $Z$, of a Stein manifold $X$ can be extended to a holomorphic function on all of $X$. Maybe this is the reason that the domain case often goes without ''name'' (just because it is a too particular case of the general theorem)