$L^{2}$ extension for jets of holomorphic sections of a hermitian line bundle

Abstract

Let $(X, \omega)$ be a weakly pseudoconvex Kähler 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 that 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. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.