### Algebraic and Real K-theory of Real Varieties, by Max Karoubi and Charles Weibel

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(C), 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(C).

Passing to finite coefficients Z/m, we show that the maps from
K_{n}(V;Z/m) to KR^{-n}(V(C);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_{C}.

When V is the affine variety of the d-sphere, it turns out that
KR^{*}(V(C))=KO^{*}(S^{d}).
In this case we show that for all nonnegative n we have

K_{n}(V;Z/m) = KO^{-n}(S^{d};Z/m).

This paper has appeared in *Topology* 42 (2003), 715-742.

Max Karoubi <karoubi@math.jussieu.fr>

Charles Weibel <weibel@math.rutgers.edu>