L²Extension for jets of holomorphic sections of a Hermitian line Bundle

Abstract

Abstract Let ( X , ω) be a weakly pseudoconvex Kähler manifold, Y ⊂ 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 > 0, any section of the jet sheaf 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.