Extending holomorphic sections from complex subvarieties. (English) Zbl 0968.32005

Forstnerič, Franc; Prezelj, Jasna

Math. Z. 236, No. 1, 43-68 (2001).

Let \(X\) be a Stein manifold and let \(Y\) be a complex manifold which admits a spray in the sense of M. Gromov [J. Am. Math. Soc. 2, No. 4, 851-897 (1989; Zbl 0686.32012)]. We prove that for every closed complex subvariety \(X_0\) of \(X\) and for every continuous map \(f_0: X\to Y\) whose restriction to \(X_0\) is holomorphic there exists a homotopy of maps \(f_t:X\to Y\) \((0\leq t\leq 1)\) whose restrictions to \(X_0\) agree with \(f_0\) and such that the map \(f_1\) 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 H. Grauert [Math. Ann. 133, 450-472 (1957; Zbl 0080.29202)] and O. Forster and K. J. Ramspott [Invent. Math. 2, 145-170 (1966; Zbl 0154.33401)].

Reviewer: F.Forstnerič (Ljubljana)

MSC:

32D15	Continuation of analytic objects in several complex variables
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results

Keywords:

Stein manifold; sections of holomorphic submersions

Citations:

Zbl 0686.32012; Zbl 0080.29202; Zbl 0154.33401