$L^2$ Extension for Jets of Holomorphic Sections of a Hermitian Line Bundle

Abstract

Let $(X, \omega)$ be a weakly pseudoconvex K\"ahler manifold, $Y \subset X$ a closed submanifold defined by some holomorphic section of a vector bundle over $X,$ and $L$ a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer $k\geq 0,$ any section of the jet sheaf $L\otimes {\cal O}_X/{\cal I}_Y^{k+1},$ which satisfies a certain $L^2$ condition, can be extended into a global holomorphic section of $L$ over $X$ whose $L^2$ growth on an arbitrary compact subset of $X$ is under control. In particular, if $Y$ is merely a point, this gives the existence of a global holomorphic function with an $L^2$ norm under control and with prescribed values for all its derivatives up to order $k$ at a point. This result generalizes the $L^2$ extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. In this respect, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.