Extending holomorphic sections from complex subvarieties

By F. Forstnerič, Jasna Prezelj. 2001

49 citations recorded by the source corpus.

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 .

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Citations in the corpus

Survey of Oka theory (2010, unknown): “Prezelj and the first-named author in [21, 22, 23].”
Holomorphic factorization of mappings into SL_n(C) (2008, background): “We need the principle in it’s strongest form suggested by Gromov, completely proven by Forstnerič and Prezelj [4], see Theorem 3.6 and also Forstnerič [3, Theorem 8.3].”
Model structures and the Oka Principle (2003, methodology): “A deep theorem of Gromov [G, FP ] implies that for any commuting square”