Joseph Ayoub

Note sur les opérations de Grothendieck et la réalisation de Betti

The purpose of this note is to show that the Betti realization of motives is compatible with Grothendiecks six operations and the nearby cycles functors, which in the motivic world, were previously studied by the author. We first review the construction of the Betti realization. Then, we establish some general criteria which, applied to the Betti realization, give the compatibilities we seek except for the one concerning the nearby cycles functors. The latter will be treated in a separate section.

Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety

We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we consider the universal Artin motive mapping to M and denote it

1-motivic sheaves and the Albanese functor

\omega^{0}_{X}(M)

Nori \(1\)-motives

. We use this to define a motive

Le Carré du Foncteur de Dualité est Monoïdal

\mathbb{E}_{X}

Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique

over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors

La réalisation étale et les opérations de Grothendieck

\omega^{0}_{X}

L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle, II

and the computation of the motives

Motifs des variétés analytiques rigides

\mathbb{E}_{X}

Une version relative de la conjecture des périodes de Kontsevich-Zagier

.In the second half of the paper, we develop their application to locally symmetric varieties. More specifically, let Γ\D be a locally symmetric variety and denote by