Ohsawa–Takegoshi extension theorem

Remarks

The point is that $C$ does not depend on $\psi$ or on $f$. The proof solves a $\bar\partial$-equation with $L^2$ estimates for a weight that is singular along $H$, using a twisted form of the Bochner–Kodaira–Morrey–Kohn identity. One extends $f$ smoothly with a cutoff in $z_n$ and corrects the extension by a solution $u$ of $\bar\partial u = g$ whose weight $e^{-\psi}/\abs{z_n}^2$ forces $u$ to vanish on $H$. The sharp value of the constant was later found by Błocki and by Guan and Zhou, and it yields the Suita conjecture on the Bergman kernel of plane domains. Example: take $\psi = 0$ and let $\Omega$ be the unit ball. Then every holomorphic $f$ on the unit ball of $\C^{n-1}$ with $\int \abs{f}^2 < \infty$ extends to a square integrable holomorphic function on the unit ball of $\C^n$. For the ball this is easy, because the trivial extension $F(z', z_n) = f(z')$ is defined on the ball and satisfies $\int_\Omega \abs{F}^2 \le \pi \int_{\Omega \cap H} \abs{f}^2$. For a general $\Omega$ the point $(z', 0)$ need not lie in $\Omega$ when $(z', z_n)$ does, so the trivial extension is not defined on $\Omega$, and the theorem is needed.

Going deeper

For a student

The theorem refines Cartan's theorem B, which gives an extension from a closed submanifold of a Stein manifold but without control of its size. Because the constant is uniform in the weight, the theorem can be applied to singular weights such as $\psi = c \log \abs{g}$. This makes it a key tool in the theory of multiplier ideals: it is used in Demailly's approximation of plurisubharmonic functions and in Siu's proof of the invariance of plurigenera.