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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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