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I. Panin proved in the nineties that the algebraic K-theory of twistedprojective homogeneous varieties can be expressed in terms of central simplealgebras. Later, Merkurjev and Panin described the algebraic K-theory of toricvarieties as a direct summand of the algebraic K-theory of separable algebras.In this article, making use of the recent theory of noncommutative motives, weextend Panin and Merkurjev-Panin computations from algebraic K-theory to everyadditive invariant. As a first application, we fully compute the cyclichomology (and all its variants) of twisted projective homogeneous varieties. Asa second application, we show that the noncommutative motive of a twistedprojective homogeneous variety is trivial if and only if the Brauer classes ofthe associated central simple algebras are trivial. Along the way we constructa fully-faithful tensor functor from Merkurjev-Panin's motivic category toKontsevich's category of noncommutative Chow motives, which is of independentinterest.

Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives