  We describe some buildings related to complex Kac-Moody groups. First we
describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn))
and its Veronese representation. Next we recall the construction of the affine
building associated to a discrete valuation on the rational function field
$C(z)$. Then we describe the same building in terms of complex Laurent
polynomials, and introduce the Veronese representation, which is an equivariant
embedding of the building into an affine Kac-Moody algebra. Next, we introduce
topological twin buildings. These buildings can be used for a proof - which is
a variant of the proof by Quillen and Mitchell - of Bott periodicity which uses
only topological geometry. At the end we indicate very briefly that the whole
process works also for affine real almost split Kac-Moody groups.
