Compact structures on \({\mathbb C}\) *\(\times {\mathbb C}\) *
type: Item
instance of: scholarly article
title: Compact structures on \({\mathbb C}\) *\(\times {\mathbb C}\) *
zbMATH Open document ID: 0628.32036
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: By a ``surface'' we mean a connected 2-dimensional complex manifold. By a `compact structure' on a surface V we mean an equivalence class of pairs (S,C) where S is a compact surface with S-C as a Zariski-open subset biholomorphic to V; two such pairs (S,C) and (S',C') being `equivalent' if there is a chain \((S,C)=(S_ 0,C_ 0),...,(S_ i,C_ i),...,(S_ n,C_ n)=(S,C')\) where each pair \((S_{i+1},C_{i+1})\) is obtained from \((S_ i,C_ i)\) by blowing down an exceptional curve of the first kind r by blowing up a point, in \(C_ i.\) We prove: Theorem A. Let V be proper homotopy equivalent \({\mathbb{C}}\times {\mathbb{C}}^ *\). Then \(({\mathbb{P}}^ 2,2L)\) is the only compact structure on it, where 2L denotes the union of two lines on \({\mathbb{P}}^ 2.\) Theorem B. Let V be proper homotopy equivalent to \({\mathbb{C}}^ *\times {\mathbb{C}}^ *\). Then the compact structures on V are one of the following: \((i)\quad ({\mathbb{P}}^ 2;3L)\) where 3L denotes the union of any three lines in general position. \((ii)\quad (X,E),\) where X is the total space of a \({\mathbb{P}}^ 1\)-bundle over an elliptic curve and E is a section with \((E^ 2)=0\). \((iii)\quad (H,E)\), where H is a Hopf surface and E an elliptic curve on H.
Mathematics Subject Classification ID: 32J05
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32Q99
zbMATH DE Number: 4021599
zbMATH Keywords: compact structure on a surface
zbMATH Keywords: blowing down
zbMATH Keywords: proper homotopy equivalent
author: Anant R. Shastri
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228076
OpenAlex ID: W2018358360
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DOI: 10.2748/TMJ/1178228076
description: scientific article; zbMATH DE number 4021599