On algebraic \(K\)-theory of real algebraic varieties with circle action
Let \(X\) be a compact, connected, orientable, nonsingular algebraic variety endowed with an algebraic free action of \(S^1\). Assume that \(Y=X/S^{1}\) is a real algebraic variety. The main result of the paper is that the projection \({\pi}: X \rightarrow Y\) induces a surjective map on reduced algebraic \(K\)-groups tensored with \(\mathbb Q\); i.e, the map \({\pi}^{*} :{\tilde K}_0({\mathcal R}(Y,{\mathbb C}))\otimes {\mathbb Q} \rightarrow {\tilde K}_0({\mathcal R}(X,{\mathbb C}))\otimes {\mathbb Q}\) is onto, where \({\mathcal R}(X,{\mathbb C})= {\mathcal R}(X)\otimes {\mathbb C}\) and \({\mathcal R}(X)\) denotes the ring of entire rational functions on the variety \(X.\) This result partially extends the Bochnak-Kucharz result that \({\tilde K}_0({\mathcal R}(X\times S^{1},{\mathbb C})) = {\tilde K}_0({\mathcal R}(X,{\mathbb C}))\) for any real algebraic variety.
scientific article; zbMATH DE number 1758364
scientific article