L² Extension for Jets of Holomorphic Sections of a Hermitian Line Bundle

2004/09/09 by Dan Popovici, Popovici, Dan
Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV

paper · pdf · doi:10.48550/arxiv.math/0409170

arxiv created 2004/09/09 · arxiv updated 2009/12/01

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 L⊗ \cal O_X/\cal I_Y^k+1, which satisfies a certain L² condition, can be extended into a global holomorphic section of L over X whose L² 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 L² norm under control and with prescribed values for all its derivatives up to order k at a point. This result generalizes the L² 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.

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