L2Extension for jets of holomorphic sections of a Hermitian line Bundle

Abstract

AbstractLet (X, ω) be a weakly pseudoconvex Kähler manifold,Y ⊂ Xa closed submanifold defined by some holomorphic section of a vector bundle overX, andLa Hermitian line bundle satisfying certain positivity conditions. We prove that for any integerk> 0, any section of the jet sheafwhich satisfies a certain L2condition, can be extended into a global holomorphic section ofLoverXwhose L2growth on an arbitrary compact subset ofXis under control. In particular, ifYis merely a point, this gives the existence of a global holomorphic function with an L2norm under control and with prescribed values for all its derivatives up to orderkat that point. This result generalizes the L2extension 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.