The equivariant Grothendieck groups of the Russell-Koras threefolds (Q2715660)

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.