Examples of algebraically realized maps
This paper discusses the following conjecture: Let \(G\) be a compact Lie group. Then every closed smooth \(G\)-manifold is equivariantly diffeomorphic to a nonsingular real algebraic \(G\)-variety \(X\) so that every \(G\) vector bundle over \(X\) is strongly algebraic, i.e., classified by an entire rational map. This algebraic realization problem can be reduced to a bordism problem. The paper under review deals with this bordism problem for the case \(G=\mathbb{Z}_4\). The authors construct examples of \(G\)-equivariant entire rational maps from non-singular real algebraic \(G\)-varieties to Grassmannians with appropriate actions. It is shown that these maps represent all classes in certain equivariant bordism groups of the Grassmannians. Using these examples one can prove the conjecture for \(G=\mathbb{Z}_4\). Moreover, in the preprint [\textit{K. H. Dovermann} and \textit{A. G. Wasserman}, ``Algebraic realization of cyclic group actions'', \url{http://www.math.hawaii.edu/~heiner/cbonn.pdf}] these examples are used to prove the above conjecture for all cyclic groups \(G\).
scientific article; zbMATH DE number 6686307
scientific article