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

Abstract

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's 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 ⊗-functor from Merkurjev–Panin's motivic category to Kontsevich's category of noncommutative Chow motives, which is of independent interest.