L 2  extension theorem for jets with variable denominators

Sheng Rao,Runze Zhang

doi:10.5802/crmath.167

Abstract

By studying the variable denominators introduced by X. Zhou–L. Zhu, we generalize the results of D. Popovici for the L 2 extension theorem for jets. As a direct corollary, we also give a generalization of T. Ohsawa–K. Takegoshi’s extension theorem to a jet version.

Highlights

Many versions and variants of the L2 extension theorems have been studied. These results lead to numerous applications in algebraic geometry and complex analysis
One interesting problem is to study the L2 extension theorem for jets
They obtained some results on weighted L2 extension of holomorphic top forms with values in a holomorphic line bundle, where the weights used are determined by the variable denominators

Summary

Introduction: main results and applications

Popovici [14], which generalized the L2 extension theorems of Ohsawa–Takegoshi– Manivel to the case of jets of sections of a line bundle over a weakly pseudoconvex Kähler manifold. Zhu [15] proved an L2 extension theorem for holomorphic sections of holomorphic line bundles equipped with singular metrics on weakly pseudoconvex Kähler manifolds. Futhermore, they obtained optimal constants corresponding to variable denominators. Let Ω ⊂ Cn be a bounded pseudoconvex open set, and Y ⊂ Ω a closed nonsingular subvariety defined by some section s ∈ H 0(Ω, E ) of a Hermitian holomorphic vector bundle E of rank m ≥ 1 with bounded curvature form.

Preliminaries for L2 extension

Proof of Theorem 1