Jacobians of Noncommutative Motives

Jacobians of Noncommutative Motives

In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a Q-linear additive Jacobian functor N 7→ J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd ...

Kontsevich’s Noncommutative Numerical Motives

In this note we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [14]. As a consequence, we conclude that NCnum(k)F is abelian semi-simple as conjectured by Kontsevich.

Noncommutative Artin Motives

In this article we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k)Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridg...

Noncommutative Motives, Numerical Equivalence, and Semi-simplicity

In this article we further the study of the relationship between pure motives and noncommutative motives, initiated in [25]. Making use of Hochschild homology, we introduce the category NNum(k)F of noncommutative numerical motives (over a base ring k and with coefficients in a field F ). We prove that NNum(k)F is abelian semi-simple and that Grothendieck’s category Num(k)Q of numerical motives ...

Noncommutative Geometry, Quantum Fields and Motives