×

zbMATH — the first resource for mathematics

Prolongement d’applications holomorphes.(Continuation of holomorphic mappings). (French) Zbl 0718.32013
Let \(T:=[(\rho \Delta)^{n-1}\times \Delta]\cup [\Delta^{n-1}\times (\Delta \setminus \tau {\bar \Delta})]\) be the Hartogs domains in the unit polydisc \(\Delta\) \({}^ n\). A complex manifold X is said to be holomorphically (resp. meromorphically) extensifer if any holomorphic mapping f: \(T\to X\) extends holomorphically (resp. meromorphically) to \(\Delta^ n\). If any holomorphic mapping f: \(T\to X\) extends holomorphically to \(\Delta^ n\setminus Z\), where Z is an analytic subset with codim \(Z\geq 2\), then X is called holomorphically extensifer outside of codimension 2.
The main results of the paper are the following two theorems. Let \(\Phi: X\to Y\) be holomorphic and let \({\mathcal U}=(U_ i)_{i\in I}\) be a covering of Y.
(1) If Y is holomorphically extensifer and if \(\Phi^{-1}(U_ i)\) is holomorphically (resp. meromorphically) extensifer for each \(i\in I\) then X is holomorphically (resp. meromorphically) extensifer.
(2) If Y is projective and if \(\Phi^{-1}(U_ i)\) is holomorphically extensifer for each \(i\in I\) then X is holomorphically extensifer outside of codimension 2.
In particular: (3) Any homogeneous manifold is holomorphically extensifer outside of codimension 2.
(4) Any compact almost homogeneous Kähler manifold is meromorphically extensifer.

MSC:
32D15 Continuation of analytic objects in several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] BARTH (W.) , PETERS (G.) and VAN DE VEN (A.) . - Compact complex surfaces . - Springer-Verlag Berlin, Heidelberg, New-York, Tokyo, 1984 . Zbl 0718.14023 · Zbl 0718.14023
[2] BENKHE (H.) und THULLEN (P.) . - Theorie der Funktionen mehrerer Komplexen Veränderlishen . - Springer-Verlag, 1970 . Zbl 0204.39502 · Zbl 0204.39502
[3] BOREL (A.) und REMMERT (R.) . - Uber kompakte homogene Kählersche Mannigfaltigkeiten , Math. Annal., t. 145, 1962 , p. 429-439. MR 26 #3088 | Zbl 0111.18001 · Zbl 0111.18001 · doi:10.1007/BF01471087 · eudml:160917
[4] DLOUSSKY (G.) . - Enveloppes d’holomorphie et prolongements d’hypersurfaces , Séminaire Pierre LELONG, p. 217-235, 1975 - 1976 , Lectures notes in Math., 578, Springer, 1977 . MR 57 #707 | Zbl 0372.32008 · Zbl 0372.32008
[5] DLOUSSKY (G.) . - Prolongements d’applications analytiques , Séminaire Pierre LELONG, Henri SKODA p. 42-95, 1976 - 1977 , Lectures notes in Math., 694, Springer, 1978 . MR 80f:32011 | Zbl 0403.32007 · Zbl 0403.32007
[6] DOCQUIER (F.) und GRAUERT (H.) . - Levisches Problem und Rungerscher Satz für Teilgebiete Steinscher Mannigfaltigkeiten , Math. Annal., t. 140, 1960 , p. 94-123. MR 26 #6435 | Zbl 0095.28004 · Zbl 0095.28004 · doi:10.1007/BF01360084 · eudml:160766
[7] GRIFFITHS (P.A.) . - Two Theorems on extensions of Holomorphic Mappings , Invent. Math., t. 14, 1971 , p. 27-62. MR 45 #2202 | Zbl 0223.32016 · Zbl 0223.32016 · doi:10.1007/BF01418742 · eudml:142104
[8] HUCKELBERRY (A.) and OELJEKLAUSS (E.) . - Classification theorems for Almost Homogeneous spaces . - Institut Elie Cartan, 1984 . Zbl 0549.32024 · Zbl 0549.32024
[9] IVASHKOVICH (S.M.) . - Extension of locally biholomorphic mappings into product of complex manifolds , Math. USSR Izvestiya, t. 27, n^\circ 1, 1986 , p. 193-199. Zbl 0595.32018 · Zbl 0595.32018 · doi:10.1070/IM1986v027n01ABEH001172
[10] IVASHKOVICH (S.M.) . - Extension of locally biholomorphic mappings of domains into complex projective space , Math. USSR Izvestiya, t. 22, n^\circ 1, 1984 , p. 181-189. Zbl 0561.32007 · Zbl 0561.32007 · doi:10.1070/IM1984v022n01ABEH001437
[11] IVASHKOVICH (S.M.) . - The Hartogs phenomen for holomorphically convex Kähler manifolds , Math. USSR Izvestiya, t. 29, n^\circ 1, 1987 , p. 225-232. Zbl 0618.32011 · Zbl 0618.32011 · doi:10.1070/IM1987v029n01ABEH000968
[12] MALGRANGE (B.) . - Lectures on the theory of fonction of several complex variables . - Tata Institute of fundamental Research Bombay, 1958 .
[13] POTTERS (J.) . - On almost homogeneous compact complex surfaces , Invent. Math., t. 8, 1969 , p. 244-266. MR 41 #3808 | Zbl 0205.25102 · Zbl 0205.25102 · doi:10.1007/BF01406077 · eudml:141988
[14] REMMERT (R.) . - Holomorphe und meromorphe abbildungen komplexer Räume , Math. Annalen, t. 133, 1957 , p. 328-370. MR 19,1193d | Zbl 0079.10201 · Zbl 0079.10201 · doi:10.1007/BF01342886 · eudml:160555
[15] WANG (H.C.) . - Complex parallelisable manifolds , Proc. AMS, t. 5, 1954 , p. 771-776. Zbl 0056.15403 · Zbl 0056.15403 · doi:10.2307/2031863
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.