Explicit generators for (conjectural) mixed motives (in Voevodsky's $\dmge$). The Kunneth decomposition of pure (numerical) motives

Abstract

In this note we describe very explicitly a rich family of mixed motives that generates Voevodsky's $D{M}_{gm}^{eff}\mathbb{Q}$ (as a triangulated category). They "should be" mixed since they have only one non-zero Betti cohomology group. Our method also allows to define a family of direct summands of the numerical motif of any smooth projective variety $P$. Modulo certain standard conjectures, this construction yields the Kunneth decomposition of the diagonal of $P$.

New very explicit mixed (conjecturally) motives in Voevodsky's $DM$!