Complex analytic compactifications of \({\mathbb{C}}\) 3
type: Item
instance of: scholarly article
title: Complex analytic compactifications of \({\mathbb{C}}\) 3
zbMATH Open document ID: 0651.14024
author: M. Furushima
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: In this note, a complete classification of smooth connected compactifications X of \({\mathbb{C}}^ 3 \)with normal divisor Y at infinity is announced. They fall into 3 classes: the 2 classical ones \((X={\mathbb{P}}_ 3({\mathbb{C}})\) or \(Q_ 3\), the 3-dimensional quadric) and a new one \((V_ 5,H_ 5)\) constructed by the author; \(V_ 5\) is a Fano 3-fold of degree \( 5\) in \({\mathbb{P}}_ 6({\mathbb{C}})\) and \(H_ 5\) a singular Del Pezzo hyperplane section of \(V_ 5\). The aim of this note is to exclude the last remaining possibility of a Fano 3-fold X of index \(r=1\). The argument uses the Iskovskij-Shokurov classification, and Mori's theory of extremal rays. For the case of \(b_ 2(X)=1\), Y non-normal, see \textit{T. Peternell} and \textit{M. Schneider}, Math. Ann. 280, No.1, 129-146 (1988; see the following review).
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 32J05
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 4061398
zbMATH Keywords: classification of smooth connected compactifications
reviewed by: Frédéric Campana
MaRDI profile type: Publication
full work available at URL: pjaa.64.25
OpenAlex ID: W3044143735
cites work: Some algebraicity criteria for singular surfaces
cites work: Singular Del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C3
cites work: Anticanonical models of three-dimensional algebraic varieties
cites work: On Minimally Elliptic Singularities
cites work: Threefolds whose canonical bundles are not numerically effective
cites work: Q5510569
DOI: 10.3792/PJAA.64.25
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description: scientific article; zbMATH DE number 4061398