  I. Panin proved in the nineties that the algebraic K-theory of twisted
projective homogeneous varieties can be expressed in terms of central simple
algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric
varieties as a direct summand of the algebraic K-theory of separable algebras.
In this article, making use of the recent theory of noncommutative motives, we
extend Panin and Merkurjev-Panin computations from algebraic K-theory to every
additive invariant. As a first application, we fully compute the cyclic
homology (and all its variants) of twisted projective homogeneous varieties. As
a second application, we show that the noncommutative motive of a twisted
projective homogeneous variety is trivial if and only if the Brauer classes of
the associated central simple algebras are trivial. Along the way we construct
a fully-faithful tensor functor from Merkurjev-Panin's motivic category to
Kontsevich's category of noncommutative Chow motives, which is of independent
interest.
