Algebraic and Real K-theory of Algebraic varieties

Let V be a smooth variety defined over the real numbers. Every algebraic vector bundle on V induces a complex vector bundle on the underlying topological space V©, and the involution coming from complex conjugation makes it a Real vector bundle in the sense of Atiyah. This association leads to a natural map from the algebraic K-theory of V to Atiyah’s ``Real K-theory’’ of V©. Passing to finite coefficients Z/m, we show that the maps from K_n(V ; Z/m) to KR ^{-n}(V©;Z/m) are isomorphisms when n is at least the dimension of V, at least when m is a power of two. Our key descent result is a comparison of the K-theory space of V with the homotopy fixed points (for complex conjugation) of the K-theory space of the complex variety V©. When V is the affine variety of the d-sphere S, it turns out that KR*(V©)=KO*(S). In this case we show that for all nonnegative n we have K_n(V ; Z/m) = KO^{-n}(S ; Z/m).