Compactification of \({\mathbb{C}}^ 3\) with reducible boundary divisor
type: Item
instance of: scholarly article
title: Compactification of \({\mathbb{C}}^ 3\) with reducible boundary divisor
zbMATH Open document ID: 0681.32020
DOI: 10.1007/BF01453582
author: Stefan Müller-Stach
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164644
review text: A compactification of \({\mathbb{C}}^ n\) is a pair (X,A), where X is a compact complex manifold and A an analytic subset, such that \(X\setminus A\) is biholomorphic to \({\mathbb{C}}^ n\). The author classifies projective compactifications of \({\mathbb{C}}^ 3\) with two normal boundary divisors at infinity intersecting transversally. They are divided in three classes: \({\mathfrak P}^ 2\)-bundles over \({\mathfrak P}^ 1\), \({\mathfrak P}^ 1\)-bundles over \({\mathfrak P}^ 2\) and blow ups of the four Fano-3-folds \({\mathfrak P}^ 3\), \({\mathfrak Q}^ 3\), \({\mathfrak V}_ 5\) and \({\mathfrak A}_{22}\), which are exactly the irreducible compactifications of \({\mathbb{C}}^ 3\). The boundary surfaces and their singularities are also classified. Especially it follows, that such a compactification is rational.
reviewed by: Stefan Müller-Stach
Mathematics Subject Classification ID: 32J05
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 4114997
zbMATH Keywords: complex projective variety
zbMATH Keywords: contraction of an extremal ray
zbMATH Keywords: Fano variety
zbMATH Keywords: singular Gorenstein surface
zbMATH Keywords: adjunction map
zbMATH Keywords: compactification of C(sup n)
zbMATH Keywords: projective compactifications of \({\mathbb{C}}^ 3\)
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4114997