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On the quasi-projectivity of compactifiable strongly pseudoconvex manifolds. (English) Zbl 1083.32010
A complex space $$X$$ is 1-convex (or strongly pseudoconvex) if $$X$$ is holomorphically convex and admits a maximally compact analytic set $$S$$, called the exceptional set; in other words $$X$$ is a modification of a Stein space in a finite number of points.
The paper under review deals with questions and examples on the embeddability into $$\mathbb{C}^m\times \mathbb{P}_n$$ and quasi-projectivity of 1-convex manifolds with (projective) exceptional set. The paper is pompously written (e.g., §1 is called “the see-saw puzzle”!) and contains many mistakes and omissions; e.g., formula (3$$'$$) on p. 507 is wrong (the author believes that a straight line and a linear hyperplane in $$\mathbb{P}^n$$ intersect in $$m> 1$$ points!); hence one of his main result, Corollary 1.7, is false. Others like Theorem 2.1 follows immediately by Grauert’s criterion on blowing-down, Theorem 2.8 is already proven by L. Alessandrini and G. Bassanelli [Ann. Inst. Fourier 51, 99–108 (2001; Zbl 0966.32008)] but the author attributes this result to himself!.
Also Lemma 2.6 is false. (Otherwise, the long cohomology sequence with supports in $$\Gamma$$ – we keep author’s notations – give the isomorphism $$H^k(M,\nu)\to H^k(M\setminus\Gamma,\nu)$$ for $$k\geq 2$$ and $$\nu$$ locally free sheaf on $$M$$: in particular $$H^n(M,\nu)= 0$$ where $$n= \dim(M)$$. But this cannot be true for all locally free sheaves $$\nu$$ on $$M$$!).

##### MSC:
 32F10 $$q$$-convexity, $$q$$-concavity 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 32J27 Compact Kähler manifolds: generalizations, classification 32T15 Strongly pseudoconvex domains 32E10 Stein spaces, Stein manifolds 32C22 Embedding of analytic spaces 32S45 Modifications; resolution of singularities (complex-analytic aspects)
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