The equivariant Grothendieck groups of the Russell-Koras threefolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 0970.14021
published in: Canadian Journal of Mathematics
publication date: 2001-05-20T00:00:00Z
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14C35
zbMATH DE Number: 1599817
zbMATH Keywords: contractible threefolds
zbMATH Keywords: equivariant Grothendieck groups
author: Jason P. Bell
MaRDI profile type: Publication
DOI: 10.4153/CJM-2001-001-X
title: The equivariant Grothendieck groups of the Russell-Koras threefolds
review text: In this article, the author computes the equivariant Grothendieck groups of the Russell-Koras threefolds. These are smooth affine contractible threefolds endowed with a hyperbolic action of \(\mathbb{C}^{*}\) such that the quotient is isomorphic to the quotient of the tangent space at the unique fixed point. Many properties of these threefolds were given by \textit{M. Koras} and \textit{P. Russell} [J. Algebr. Geom. 6, 671-695 (1997; Zbl 0882.14013)]. In particular, they are hypersurfaces in \(\mathbb{C}^{4}\) given in two different ways. It has been shown by \textit{N. Mohan Kumar} and \textit{M. P. Murthy} that all vector bundles over such a threefold are stably free [see Ann. Math., II. Ser. 116, 579-591 (1982; Zbl 0519.14009)]. NEWLINENEWLINENEWLINEIn the present article, the equivariant setting with respect to the given action of \(\mathbb{C}^{*}\) is studied. The equivariant Grothendieck groups are computed. In particular, it is shown that if \(X\) is not isomorphic to \(\mathbb{C}^{3}\), then there are equivariant vector bundles over \(X\) which are not stably trivial. This is in constrast to the case of equivariant vector bundles over representation spaces of \(\mathbb{C}^{*}\), which are known to be not only stably trivial but in fact trivial.
reviewed by: Lucy Moser-Jauslin
description: scientific article; zbMATH DE number 1599817