Extending holomorphic sections from complex subvarieties

2001/01/04 by Franc Forstneric, Jasna Prezelj
Mathematics · #math.CV #msc:32L05 #msc:32D15 #msc:32E10 #msc:32F32

paper · pdf · doi:10.1007/s002090000169

published as Math. Z. 236 (2001), 43-68 · 23 pages

arxiv created 2001/01/04 · arxiv updated 2009/11/30

Abstract

Let X be a Stein manifold and let Y be a complex manifold which admits a spray in the sense of Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, pp. 851-897 (1989)). We prove that for every closed complex subvariety X₀ of X and for every continuous map f₀ from X to Y whose restriction to X₀ is holomorphic there exists a homotopy of maps f_t from X to Y whose restrictions to X₀ agree with f₀ and such that the map f₁ is holomorphic on X. We obtain analogous results for sections of holomorphic submersions with sprays over Stein manifolds or Stein spaces. Our results extend those of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, pp. 450-472 (1957)) and Forster and Ramspott (Analytische Modulgarben und Endromisbundel, Invent. Math. 2, pp. 145-170 (1966)).

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