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On the embedding of 1-convex manifolds with 1-dimensional exceptional set. (English) Zbl 0966.32008
Recall that a 1-convex (also called a strongly pseudoconvex) complex space $$X$$ is a proper modification $$f:X\to Y$$ at finitely many points of a Stein space $$Y$$. The union of all positive dimensional compact subspaces of $$X$$ is the exceptional set of $$X$$. A 1-convex space is called embeddable if it is biholomorphic to a closed analytic subspace of $$\mathbb{C}^N\times \mathbb{C}\mathbb{P}^m$$ for some $$N$$, $$m$$. Vo Van Tan and M. Coltoiu [see M. Coltoiu, Rev. Roum. Math. Pures Appl. 43, No. 1-2, 97-104 (1998; Zbl 0932.32018)] proved that every 1-convex manifold with 1-dimensional exceptional set $$S$$ is embeddable, except possibly if $$S$$ contains $$\mathbb{C}\mathbb{P}^1$$ with three possible normal bundles and in each of these cases there are examples of non-embeddable 1-convex manifolds. In the paper under review the authors prove the following topological criteria for the embeddability of $$X$$.
Theorem 1. Let $$X$$ be a 1-convex manifold with 1-dimensional exceptional set $$S$$. $$X$$ is Kähler if and only if $$S$$ does not contain any effective curve $$C$$ which is a boundary, i.e. such that $$H_2(X,\mathbb{Z})= 0$$.
Theorem 2. Let $$X$$ be a 1-convex manifold with 1-dimensional exceptional set $$S$$. If $$H_2(X,\mathbb{Z})$$ is finitely generated, then $$X$$ is embeddable if and only if it is Kähler.

##### MSC:
 32F10 $$q$$-convexity, $$q$$-concavity 53B35 Local differential geometry of Hermitian and Kählerian structures 32J99 Compact analytic spaces 32C22 Embedding of analytic spaces
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##### References:
  Metric properties of manifolds bimeromorphic to compact Kähler spaces, J. Differential Geom., 37, 95-121, (1993) · Zbl 0793.53068  Sur LES fibres infinitésimales d’un morphisme propre d’espaces complexes, Sém. F. Norguet, Fonctions de plusieurs variables complexes IV, 807, (1980), Springer · Zbl 0445.32019  A cut-off theorem for plurisubharmonic currents, Forum Math., 6, 567-595, (1994) · Zbl 0808.32010  On the embedding of 1-convex manifolds with 1-dimensional exceptional set, Comment. Math. Helv., 60, 458-465, (1985) · Zbl 0583.32047  On 1-convex manifolds with 1-dimensional exceptional set (collection of papers in memory of martin jurchescu), Rev. Roum. Math. Pures Appl., 43, 97-104, (1998) · Zbl 0932.32018  On the Oka-grauert principle for 1-convex manifolds, Math. Ann., 310, 3, 561-569, (1998) · Zbl 0902.32011  On hulls of meromorphy and a class of Stein manifolds, Ann. Scuola Norm. Sup., XXVIII, 405-412, (1999) · Zbl 0952.32008  On strongly q-pseudoconvex spaces with positive vector bundles, Mem. Fac. Sci. Kyushu Univ. Ser. A, 28, 135-146, (1974) · Zbl 0297.32012  An intrinsec characterization of Kähler manifolds, Invent. Math., 74, 169-198, (1983) · Zbl 0553.32008  On the existence of special metrics in complex geometry, Acta Math., 143, 261-295, (1983) · Zbl 0531.53053  The Levi problem for complex spaces II, Math. Ann., 146, 195-216, (1962) · Zbl 0131.30801  Topological Vector Spaces, 3, (1970), Springer · Zbl 0217.16002  Familien negativer vektorbündel und 1-convexe abbilungen, Abh. Math. Sem. Univ. Hamburg, 47, 150-170, (1978) · Zbl 0391.32011  Embedding strongly $$(1,1)$$-convex-concave spaces in $${\Bbb C}^m×{\Bbb C}{\Bbb P}_n,$$ Several complex variables, Vol. XXX, Part 2, 41-44, (1977), Amer. Math. Soc., Providence R.I. · Zbl 0366.32006  Embedding theorems and Kählerity for 1-convex spaces, Comment. Math. Helv., 57, 196-201, (1982) · Zbl 0555.32012  On the Kählerian geometry of 1-convex threefolds, Forum Math., 7, 131-146, (1995) · Zbl 0839.32003
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