In a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007)

In this article, we give an unconditional construction of a motivic analogue of the intermediate extension in the context of Chow motives of Abelian type. Our main application concerns intermediate extensions of Chow motives associated to Kuga families to the Baily-Borel compactification of a Shimura variety.

We introduce the notion of the boundary motive of a scheme X over a perfect field. By definition, it measures the difference between the motive M gm (X) and the motive with compact support M c gm (X), as defined and studied by Voevodsky et al. in Cycles, transfers, and motivic homology theories (Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)). We develop three tools to compute the boundary motive in terms of the geometry of a compactification of X: co-localization; invariance under abstract blow-up; and analytical invariance. We then prove auto-duality of the boundary motive of a smooth scheme X. As a formal consequence of this, and of co-localization, we obtain a fourth computational tool, namely localization for the boundary motive. In a sequel to this work (J. Wildeshaus, On the boundary motive of a Shimura variety, Prépublications du Laboratoire d'Analyse, Géométrie et Applications de l'Université Paris 13, no. 2004-23), these tools will be applied to Shimura varieties.