\(R\)-equivalence and special unitary groups
The notion of \(R\)-equivalence, introduced by \textit{Yu. I. Manin} [``Cubic forms: Algebra, geometry, arithmetic'' (1972; Zbl 0255.14002; English translation 1974; Zbl 0277.14014)], is an important birational invariant of an algebraic variety defined over an arbitrary field \(F\). Let \(G\) be an algebraic group over an arbitrary field \(F\). The set of \(R\)-equivalence classes \(G(F)/R\), has a natural group structure. If for instance the group \(G\) is connected and a direct factor of a stably rational group, then the group of \(R\)-equivalence classes of \(G\) is trivial. But the converse is known only for tori. However, the group of \(R\)-equivalence classes carries some information about the birational geometry of \(G\). For any simply connected semisimple classical algebraic group \(G\), a norm homomorphism is constructed and for all the special unitary groups the group of \(R\)-equivalence classes is calculated.
scientific article; zbMATH DE number 1234247
scientific article