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Zbl 1095.32006
Chiose, Ionuţ
On the embedding and compactification of $q$-complete manifolds. (English)
[J] Ann. Inst. Fourier 56, No. 2, 373-396 (2006). ISSN 0373-0956; ISSN 1777-5310/e

If $X$ is a connected pseudoconcave complex manifold of dimension $n$ and if $q \geq 2$, the author furnishes some sufficient conditions in order to compactify $X$. Almost all conditions are also necessary, while asking that $X$ be $(n-q+1)$-concave is ``generically'' necessary. This result is an ``interpolation'' result between the case where one can compactify $X$ by adding a hyperplane at infinity and the case where the compactification can be done by adding finitely many points. \par Next, the author furnishes necessary and sufficient conditions for the connected complex manifold $X$ of dimension $n$ to be biholomorphic to a proper submanifold of ${\Bbb {P}}^{1} \times {\Bbb{C}}^{N}$. As a consequence one obtains a characterization of a special class of holomorphically convex manifolds which can be embedded into ${\Bbb {P}}^{N} \backslash {\Bbb {P}}^{N-2}$. Thus the author gives some special answers to the problem of characterizing intrinsically the proper submanifolds of ${\Bbb {P}}^{N} \backslash {\Bbb {P}}^{N-q}$ raised in the paper [{\it F. Harvey} and {\it H. Lawson}, Ann. Math. (2) 106, No. 2, 213--238 (1977; Zbl 0361.32010)].
[Eugen Pascu (Montréal)]

MSC 2000:: *32Q40 Embedding theorems
32J05 Compactification of analytic spaces
32F10 $q$-convexity, etc.
32L15 Bundle convexity

Keywords: pseudoconvex space; pseudoconcave space; embedding; compactification; positive line bundles; Remmert reduction

Citations: Zbl 0361.32010

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