A tribute to a work of C. P. Ramanujam
type: Item
instance of: scholarly article
title: A tribute to a work of C. P. Ramanujam
zbMATH Open document ID: 0694.14014
DOI: 10.1007/BF02863636
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1988-01-01T00:00:00Z
review text: Let V be a smooth affine surface with coordinate ring \(\Gamma\) (V) a U.F.D. This happens for example if V is contractible. Then: \(V\cong {\mathbb{C}}^ 2\) if either: (1) \(\pi_ 1^{\infty}(V)\), the fundamental group at infinity, is trivial, or: (2) \({\bar \kappa}\)(V)\(=-\infty\), where \({\bar \kappa}\):\(=\log arithmic\) Kodaira dimension. Ramanujam (essentially) proved (1) by considering the fundamental group of the real 3-dimensional boundary \(\partial T\) of a tubular neighborhood of the divisor at infinity of a compactification \(\bar V\) of V. Fujita, Miyanishi and Sugie proved (2) by purely geometrical arguments resting on the existence of Zariski decomposition of pseudo-effective divisors. This expository article describes these approaches, as well as the direct proof of the relationship: (2)\(\Rightarrow (1)\) (the author). The existence of V with \({\bar \kappa}(V)=1\quad or\quad 2\) which are homology cells (the author and Miyanishi), as well as the rationality of these (the author and Shastri) are related questions.
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 4136109
zbMATH Keywords: characterization of complex plane
zbMATH Keywords: fundamental group
zbMATH Keywords: logarithmic Kodaira dimension
zbMATH Keywords: rationality
author: Rajendra Gurjar
reviewed by: Frédéric Campana
MaRDI profile type: Publication
cites work: On Zariski problem
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full work available at URL: https://doi.org/10.1007/bf02863636
OpenAlex ID: W2025954028
description: scientific article; zbMATH DE number 4136109