On algebraic K-theory of real algebraic varieties with circle action

Abstract

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S 1-action so that the quotient Y= X/ S 1 is also a real algebraic variety. If π : X→Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with Q , π ∗ : K ̃ 0( R(Y, C))⊗ Q→ K ̃ 0( R(X, C))⊗ Q is onto, where R(X, C)= R(X)⊗ C , R(X) denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that K ̃ 0( R(X×S 1, C))= K ̃ 0( R(X, C)) for any real algebraic variety X. As an application we will show that for a compact connected Lie group G K ̃ 0( R(G, C))⊗ Q=0 .