### Abstract:

On a complex variety X, two different approaches to K-theory are available: the
algebraic K-theory of the variety, and the topological K-theory of the underlying
topological space. In this context, the algebraic variant known as Hermitian
K-theory corresponds to topological KO-theory. Our aim is to compare the two
approaches.
We start by constructing a comparison map from certain Hermitian K-groups
of X to the KO-groups of X. It is clear what this map must be on groups in
degree zero, but the definitions of relative and higher groups differ widely in
the algebraic and the topological setting. This difficulty can be overcome by
viewing relative and higher groups as subgroups of degree zero groups of certain
auxiliary spaces.
Once the definition of our comparison map is in place, we prove a number
of fundamental properties, in particular compatibility with pushforwards along
closed embeddings. We also show how we can use it to compare an exact
sequence relating usual algebraic K-theory to Hermitian K-theory with a portion
of the Bott sequence in topology. This finally allows us to deduce that the
map is an isomorphism on smooth cellular varieties. We conclude with some
details concerning projective spaces, for which independent computations of the
algebraic and the topological groups exist.