Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives

Mixed Motives

2.1 Weight-two complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Beilinson-Lichtenbaum complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Bloch’s cycle complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Suslin homology and Friedlander-Suslin cohomology . . . . . . . . . . . . . 17 2.5 C...

Periods and mixed motives

We define motivic multiple polylogarithms and prove the double shuffle relations for them. We use this to study the motivic fundamental group π 1 (Gm −μN ), where μN is the group of all N -th roots of unity, and relate the structure of π 1 (Gm − μN ) to the geometry and topology of modular varieties Xm(N) := Γ1(m;N)\GLm(R)/R ∗ + ·Om for m = 1, 2, 3, 4, .... We get new results about the action o...

Mixed elliptic motives

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.

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 ...