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

TL;DR

This paper introduces Artin motives and constructs a motive E_X to analyze singularities, applying these concepts to locally symmetric varieties and relating their cohomology to motives via compactifications.

Contribution

It develops the theory of Artin and cohomological motives over schemes and applies it to the study of locally symmetric varieties and their compactifications.

Findings

The motive E_X captures singularity invariants of scheme X.

The pushforward of the motive E_Z corresponds to the cohomology of the compactification W.

The Betti realization of E_Z matches the cohomology of the locally symmetric variety W.

Abstract

We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its punctual weight zero part $\omega^0_X(M)$ 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. The first half of the paper is devoted to the study of the functors $\omega^0_X$ and the computation of the motives E_X. In the second half of the paper, we develop the application to locally symmetric varieties. Specifically, let Y be a locally symmetric variety and denote by p:W-->Z the projection of its reductive Borel-Serre compactification W onto its Baily-Borel Satake compactification Z. We show that $Rp_*(\Q_W)$ is naturally isomorphic to the Betti realization of the motive E_Z, where Z is viewed as a scheme. In particular, the direct image of E_Z along the projection of Z to Spec(C) gives a motive whose Betti realization is naturally isomorphic to the cohomology of W.