Algebraic structures on certain 3-folds
type: Item
instance of: scholarly article
title: Algebraic structures on certain 3-folds
zbMATH Open document ID: 0594.14032
DOI: 10.1007/BF01458021
author: Thomas Peternell
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164091
review text: This paper is the second part of a previous one [the author, Manuscr. Math. 50, 397-428 (1985; Zbl 0573.32027)] and is devoted to complete the proof of the following theorem (filling a gap pointed out by the author in an ''addendum'' to the previous paper). Let \(X\) be a Moishezon 3-fold which is topologically isomorphic to \(\mathbb{P}^3\); then \(X\) is biholomorphic to \(\mathbb{P}^3\). The analogous statement for the smooth 3- dimensional quadric is proved simultaneously. Let \(X\) be as above; assuming that the theorem is not true, the author takes a modification \(\pi : \hat X\to X,\) which is a finite sequence of blow-ups with smooth centers, such that \(\hat X\) is projective. Then he studies the extremal rays on \(\hat X\) and the corresponding modifications according to Mori's theory [\textit{S. Mori}, Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)] finding a contradiction, after a case by case analysis.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 32H99
zbMATH DE Number: 3955113
zbMATH Keywords: characterization of complex projective 3-space
zbMATH Keywords: Moishezon 3-fold
zbMATH Keywords: extremal rays
zbMATH Keywords: modifications
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3955113