Florian Ivorra: The Artin part of a motive, weights and motivic nearby sheaves

Abstract: The notion of Artin part of a relative cohomological motive has been introduced by Ayoub and Zucker in their study of motives associated with compactifications of arithmetic quotients of Hermitian symmetric spaces. They predict the existence of a relation between the Artin part of relative motive and the (conjectural) punctual weight filtration on the heart of the (conjectural) motivic t-structure on the triangulated category of motives. 

In this talk, I will present a recent joint work with Julien Sebag in which verify the Hodge theoretic side of Ayoub-Zucker’s prediction for smooth cohomological motives. I will explain how to compute the Artin part of the motivic nearby sheaf and relate it to the Betti cohomology of Berkovich spaces defines by tubes in non-Archimedean geometry. In the first part of the lecture, I will explain the results about motivic nearby sheaves and their relation with tubes needed and obtained in a previous joint work with Ayoub and Sebag.