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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)
##### Keywords:
holomorhic extension; meromorphic extension
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##### References:
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