Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety, by Joseph Ayoub and Steven Zucker

We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its weight-zero part as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. We then give an application to locally symmetric varieties. Namely, we prove that the Betti realization of E_X for X the Baily-Borel compactification is isomorphic to the cohomological direct image of the constant sheaf Q along the projection from the reductive Borel-Serre compactification to the Baily-Borel compactification.

Joseph Ayoub <>
Steven Zucker <>