Extending holomorphic sections from complex subvarieties

Abstract

In this paper we treat the classical problem of extending holomorphic mappings and sections from closed complex subvarieties of a complexmanifold. Our main results (Theorems 1.1 and 1.4) extend those of Grauert [Gr2] and Cartan [Car]; results of this type are commonly referred to as the ‘OkaGrauert principle’ on Stein manifolds. Our methods are similar to those developed in the papers of Gromov [Gro], Henkin and Leiterer [HL2] and the authors [FP1], [FP2]. The main addition here is the interpolation of a given holomorphic section on a complex subvariety of a Stein manifold. Our globalization scheme follows very closely the one developed in [FP2]. To state our first result we recall the notion of a (dominating) spray introduced by M. Gromov ([Gro], Sect. 0.5). Given a holomorphic vector bundle p:E → Y over a complex manifold Y , we denote by 0y ∈ Ey = p−1(y) the zero element in the fiberEy and we observe thatEy is aC-linear subspace of the tangent space T0yE. Definition 1. A spray on a complex manifold Y is a holomorphic vector bundle p:E → Y , together with a holomorphic map s:E → Y , such that for each y ∈ Y , s(0y) = y and the derivative ds:T0yE → TyY maps Ey surjectively onto TyY .